The document provides information about radicals, exponents, and equations for an exam. It defines square roots, even roots, cube roots, and odd roots. It explains that the square root of a negative number does not exist in the real number system. Radical expressions are defined in terms of their index, radical sign, and radicand. Rational and irrational radical expressions are also discussed. The document also defines exponential expressions and their bases and exponents. Rules are provided for negative exponents, quotient rule, and rational exponents.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.
This document provides information about radicals and working with radical expressions. It defines square roots, principal and negative square roots, radicands, perfect squares, cube roots, nth roots, and the product, quotient, and power rules for radicals. It discusses simplifying radical expressions using these rules as well as adding, subtracting, multiplying, and dividing radicals. The document also covers rationalizing denominators, solving radical equations, and using the Pythagorean theorem and distance formula.
Little Red Riding Hood needs to walk 5 miles to get to her destination. The document shows the time it would take her to walk 5 miles at different speeds: 5 hours if walking at 1 mile per hour, 1 hour if walking at 5 miles per hour, and 0.5 hours or 30 minutes if running at 10 miles per hour. The algebra expression that calculates time for any speed is 5 divided by the speed (5/S). Word problems are translated into algebra expressions so they can be solved mathematically. Examples of key words that translate into different algebra operations are provided.
Simplification of Fractions and Operations on FractionsVer Louie Gautani
The document discusses various operations involving fractions, including simplifying, converting between mixed and improper fractions, multiplying, dividing, adding, and subtracting fractions. It provides examples of performing each operation step-by-step and simplifying the resulting fraction. Rules for working with fractions are reviewed and examples of applying the rules are shown.
The document discusses factoring the difference of two squares. It involves reviewing factoring the difference of two squares, which involves recognizing that the difference of two squares can be written as the product of two binomials, where one binomial contains the sum of the two terms and the other contains their difference.
This document discusses one-to-one functions and their inverses. It defines a one-to-one function as a function where no two x-values are mapped to the same y-value. The inverse of a one-to-one function f is defined as f^-1 where the inputs and outputs are swapped. Examples are provided of finding the inverse of various functions by swapping variables and solving for y in terms of x. The domain of an inverse function is the range of the original function, and the range of the inverse is the domain of the original function.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
This document provides instructions for factoring trinomials with leading coefficients of 1 or greater than 1. For trinomials with a leading coefficient of 1, the document explains how to list the factors of the last term, identify the factor pair that sums to the middle term, and write the factors. For trinomials with a leading coefficient greater than 1, the instructions are to find the product of the leading and last terms, identify factor pairs that sum to the middle term, rewrite the trinomial, group terms, and factor. Examples are provided to demonstrate the process.
This document provides information about radicals and working with radical expressions. It defines square roots, principal and negative square roots, radicands, perfect squares, cube roots, nth roots, and the product, quotient, and power rules for radicals. It discusses simplifying radical expressions using these rules as well as adding, subtracting, multiplying, and dividing radicals. The document also covers rationalizing denominators, solving radical equations, and using the Pythagorean theorem and distance formula.
Little Red Riding Hood needs to walk 5 miles to get to her destination. The document shows the time it would take her to walk 5 miles at different speeds: 5 hours if walking at 1 mile per hour, 1 hour if walking at 5 miles per hour, and 0.5 hours or 30 minutes if running at 10 miles per hour. The algebra expression that calculates time for any speed is 5 divided by the speed (5/S). Word problems are translated into algebra expressions so they can be solved mathematically. Examples of key words that translate into different algebra operations are provided.
Simplification of Fractions and Operations on FractionsVer Louie Gautani
The document discusses various operations involving fractions, including simplifying, converting between mixed and improper fractions, multiplying, dividing, adding, and subtracting fractions. It provides examples of performing each operation step-by-step and simplifying the resulting fraction. Rules for working with fractions are reviewed and examples of applying the rules are shown.
The document discusses factoring the difference of two squares. It involves reviewing factoring the difference of two squares, which involves recognizing that the difference of two squares can be written as the product of two binomials, where one binomial contains the sum of the two terms and the other contains their difference.
This document discusses one-to-one functions and their inverses. It defines a one-to-one function as a function where no two x-values are mapped to the same y-value. The inverse of a one-to-one function f is defined as f^-1 where the inputs and outputs are swapped. Examples are provided of finding the inverse of various functions by swapping variables and solving for y in terms of x. The domain of an inverse function is the range of the original function, and the range of the inverse is the domain of the original function.
This document discusses linear relations and how to represent relationships between two variables with tables, graphs, and equations. It explains that a linear relation produces a straight line when graphed and can be represented by an equation of the form y = mx + b, where m is the slope and b is the y-intercept. Examples are provided to illustrate how to determine the equation from a table of values and vice versa, as well as how to graph lines from equations or data tables.
Graphing Quadratic Functions in Standard Formcmorgancavo
This document discusses graphing quadratic functions of the form y = ax^2 + bx + c. It provides the following key points:
- Quadratic functions produce parabolic graphs that open up or down depending on whether a is positive or negative.
- The vertex of the parabola is the point of minimum or maximum, which corresponds to the line of symmetry that passes through it.
- To graph a quadratic, one finds the line of symmetry, determines the vertex coordinates, and plots at least four other points to connect into a smooth curve.
Here are the steps to solve each inequality and graph the solution:
1) m + 14 < 4
-14 -14
m < -10
Graph: m < -10
2) -7 > y-1
+1 +1
-6 > y
y < -6
Graph: y < -6
3) (-3)k < 10(-3)
-3
k > -30
Graph: k > -30
4) 2x + 5 ≤ x + 1
-x -x
x + 5 ≤ 1
-5 -5
x ≤ -4
Graph: x ≤ -4
The document defines multiples as the result of multiplying two numbers. It provides examples of listing the multiples of various numbers like 2, 5, 4, and 13. It includes exercises for students to identify multiples and write multiples between given numbers. The document also introduces the concept of consecutive multiples as multiples that follow one another numerically.
This document discusses properties of logarithms, including:
1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents.
2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm.
3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.
John Napier, a Scottish mathematician and astronomer, discovered logarithms in the late 16th century as a way to simplify calculations. He introduced the concept of logarithms to ease complex mathematical computations. Napier was also an astrologer and believer in black magic who would travel with a spider and black rooster he claimed were his familiars.
This document provides instruction on ratios, proportions, and solving proportions with variables. It begins with defining ratios as comparisons between two sets of numbers and provides examples of common ratios like miles per hour. It then discusses the different ways to write ratios, such as using "to", a colon, or as a fraction. The document also covers reducing ratios, determining if two ratios form a proportion by cross-multiplying, and using proportions to solve for unknown values. Examples are provided to demonstrate setting up and solving proportions step-by-step.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
Graphing Linear Inequalities in Two Variables.pptxNadineThomas4
This document provides instructions for graphing linear inequalities in two variables on a coordinate plane. It explains that dashed lines are used for inequalities with > or < signs, while solid lines are used for >= or <= signs. Examples are given for graphing various inequalities, including graphing lines involving both x and y variables and shading the correct region. The document concludes with an example word problem involving representing having less than $5 in coins with an inequality and graphing the solution.
This document discusses direct and inverse variations. It provides examples of how to set up and solve direct and inverse variation problems. For direct variation, as one variable increases, the other increases at a constant rate. The formula is y=kx, where k is the constant rate of change. For inverse variation, as one variable increases, the other decreases. To solve inverse problems, the formula used is x1y1=x2y2, where x1 and y1 are the known values and x2 is the unknown value being solved for. Examples of setting up and solving both direct and inverse variation word problems are provided.
The document discusses ratios and provides examples using Lucky Charms cereal. It states there are 287 marshmallow pieces and 2,583 oat pieces in one box of Lucky Charms. This ratio of marshmallows to oats can be written in three ways: as a fraction, using the word "to", or using a colon. The document also discusses writing ratios in simplest form and explaining their meanings.
This document discusses how to solve absolute value inequalities by:
1) Determining whether the absolute value is greater than or less than the variable, which indicates a disjunction or conjunction graph.
2) Solving the inequality for both possibilities of the expression inside the absolute value being positive or negative.
3) Combining the solutions from both possibilities using the appropriate inequality symbol (>, <, etc.) to obtain the final solution set.
This document discusses working with rational expressions, including:
1) Finding the numbers that must be excluded from the domain to avoid undefined expressions.
2) Simplifying rational expressions by factoring numerators and denominators and cancelling common factors.
3) Multiplying, dividing, adding, and subtracting rational expressions by finding common denominators.
The document discusses factoring perfect square trinomials (polynomials with three terms where the first and last terms are perfect squares). It provides examples of factoring expressions like x^2 + 8x + 16 into (x + 4)^2. For an expression to be a perfect square trinomial, the first term must be a perfect square, the third term must be a perfect square, and the middle term must be twice the product of the square roots of the first and last terms. Students are provided examples and exercises to practice factoring various square trinomial expressions.
This document discusses direct and inverse variations. Direct variation means that as one variable increases, the other increases or decreases at a constant rate. The relationship can be expressed as y1/x1 = y2/x2. Inverse variation means that as one variable increases, the other decreases. The relationship is expressed as x1y1 = x2y2. Examples are provided of setting up and solving direct and inverse variation problems to find unknown variables.
Solving quadratic equations using the quadratic formulaDaisyListening
The document provides instructions on solving quadratic equations using the quadratic formula. It begins by presenting the formula: x = -b ± √(b^2 - 4ac) / 2a. It then works through three examples of applying the formula step-by-step: solving 2x^2 - 5x - 3 = 0, solving 2x^2 + 7x = 9, and solving x^2 + x - 1 = 0. Each example shows identifying the a, b, and c coefficients and plugging them into the formula to solve for x.
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
The document introduces the quotient rule for taking the derivative of functions that are divided. Specifically, it states that if h(x) is defined as u(x)/v(x), then the quotient rule says that h'(x) is equal to (v(x)u'(x) - u(x)v'(x))/(v(x))^2. This rule, which rearranges the fraction and applies the product rule, was developed by Maria Gaetana Agnesi in her 1748 textbook as a way to help her brothers learn algebra. It then provides two examples of using the quotient rule to find the gradient function.
This document discusses linear relations and how to represent relationships between two variables with tables, graphs, and equations. It explains that a linear relation produces a straight line when graphed and can be represented by an equation of the form y = mx + b, where m is the slope and b is the y-intercept. Examples are provided to illustrate how to determine the equation from a table of values and vice versa, as well as how to graph lines from equations or data tables.
Graphing Quadratic Functions in Standard Formcmorgancavo
This document discusses graphing quadratic functions of the form y = ax^2 + bx + c. It provides the following key points:
- Quadratic functions produce parabolic graphs that open up or down depending on whether a is positive or negative.
- The vertex of the parabola is the point of minimum or maximum, which corresponds to the line of symmetry that passes through it.
- To graph a quadratic, one finds the line of symmetry, determines the vertex coordinates, and plots at least four other points to connect into a smooth curve.
Here are the steps to solve each inequality and graph the solution:
1) m + 14 < 4
-14 -14
m < -10
Graph: m < -10
2) -7 > y-1
+1 +1
-6 > y
y < -6
Graph: y < -6
3) (-3)k < 10(-3)
-3
k > -30
Graph: k > -30
4) 2x + 5 ≤ x + 1
-x -x
x + 5 ≤ 1
-5 -5
x ≤ -4
Graph: x ≤ -4
The document defines multiples as the result of multiplying two numbers. It provides examples of listing the multiples of various numbers like 2, 5, 4, and 13. It includes exercises for students to identify multiples and write multiples between given numbers. The document also introduces the concept of consecutive multiples as multiples that follow one another numerically.
This document discusses properties of logarithms, including:
1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents.
2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm.
3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.
John Napier, a Scottish mathematician and astronomer, discovered logarithms in the late 16th century as a way to simplify calculations. He introduced the concept of logarithms to ease complex mathematical computations. Napier was also an astrologer and believer in black magic who would travel with a spider and black rooster he claimed were his familiars.
This document provides instruction on ratios, proportions, and solving proportions with variables. It begins with defining ratios as comparisons between two sets of numbers and provides examples of common ratios like miles per hour. It then discusses the different ways to write ratios, such as using "to", a colon, or as a fraction. The document also covers reducing ratios, determining if two ratios form a proportion by cross-multiplying, and using proportions to solve for unknown values. Examples are provided to demonstrate setting up and solving proportions step-by-step.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
Graphing Linear Inequalities in Two Variables.pptxNadineThomas4
This document provides instructions for graphing linear inequalities in two variables on a coordinate plane. It explains that dashed lines are used for inequalities with > or < signs, while solid lines are used for >= or <= signs. Examples are given for graphing various inequalities, including graphing lines involving both x and y variables and shading the correct region. The document concludes with an example word problem involving representing having less than $5 in coins with an inequality and graphing the solution.
This document discusses direct and inverse variations. It provides examples of how to set up and solve direct and inverse variation problems. For direct variation, as one variable increases, the other increases at a constant rate. The formula is y=kx, where k is the constant rate of change. For inverse variation, as one variable increases, the other decreases. To solve inverse problems, the formula used is x1y1=x2y2, where x1 and y1 are the known values and x2 is the unknown value being solved for. Examples of setting up and solving both direct and inverse variation word problems are provided.
The document discusses ratios and provides examples using Lucky Charms cereal. It states there are 287 marshmallow pieces and 2,583 oat pieces in one box of Lucky Charms. This ratio of marshmallows to oats can be written in three ways: as a fraction, using the word "to", or using a colon. The document also discusses writing ratios in simplest form and explaining their meanings.
This document discusses how to solve absolute value inequalities by:
1) Determining whether the absolute value is greater than or less than the variable, which indicates a disjunction or conjunction graph.
2) Solving the inequality for both possibilities of the expression inside the absolute value being positive or negative.
3) Combining the solutions from both possibilities using the appropriate inequality symbol (>, <, etc.) to obtain the final solution set.
This document discusses working with rational expressions, including:
1) Finding the numbers that must be excluded from the domain to avoid undefined expressions.
2) Simplifying rational expressions by factoring numerators and denominators and cancelling common factors.
3) Multiplying, dividing, adding, and subtracting rational expressions by finding common denominators.
The document discusses factoring perfect square trinomials (polynomials with three terms where the first and last terms are perfect squares). It provides examples of factoring expressions like x^2 + 8x + 16 into (x + 4)^2. For an expression to be a perfect square trinomial, the first term must be a perfect square, the third term must be a perfect square, and the middle term must be twice the product of the square roots of the first and last terms. Students are provided examples and exercises to practice factoring various square trinomial expressions.
This document discusses direct and inverse variations. Direct variation means that as one variable increases, the other increases or decreases at a constant rate. The relationship can be expressed as y1/x1 = y2/x2. Inverse variation means that as one variable increases, the other decreases. The relationship is expressed as x1y1 = x2y2. Examples are provided of setting up and solving direct and inverse variation problems to find unknown variables.
Solving quadratic equations using the quadratic formulaDaisyListening
The document provides instructions on solving quadratic equations using the quadratic formula. It begins by presenting the formula: x = -b ± √(b^2 - 4ac) / 2a. It then works through three examples of applying the formula step-by-step: solving 2x^2 - 5x - 3 = 0, solving 2x^2 + 7x = 9, and solving x^2 + x - 1 = 0. Each example shows identifying the a, b, and c coefficients and plugging them into the formula to solve for x.
1. A function is a relation where each input is paired with exactly one output.
2. To determine if a relation is a function, use the vertical line test - if any vertical line intersects more than one point, it is not a function.
3. To find the value of a function, substitute the given value for x into the function equation and simplify.
The document introduces the quotient rule for taking the derivative of functions that are divided. Specifically, it states that if h(x) is defined as u(x)/v(x), then the quotient rule says that h'(x) is equal to (v(x)u'(x) - u(x)v'(x))/(v(x))^2. This rule, which rearranges the fraction and applies the product rule, was developed by Maria Gaetana Agnesi in her 1748 textbook as a way to help her brothers learn algebra. It then provides two examples of using the quotient rule to find the gradient function.
This document discusses rules for taking derivatives of various functions including:
1. The derivative of a constant function is 0.
2. The power rule states that the derivative of x^n is nx^{n-1}.
3. Higher derivatives can be found by taking additional derivatives, and the nth derivative is written as f^(n).
It also covers the product rule, quotient rule, and applying rules to polynomials and exponential functions.
This document discusses basic differentiation rules including the constant rule, power rule, constant multiple rule, sum and difference rules, and derivatives of sine and cosine functions. These rules define how to take the derivative of simple functions and are fundamental to calculus.
The document provides guidance on how to write an effective introduction for an essay. It explains that an introduction should indicate the topic, describe the structure of the essay, and state the thesis. The document compares two sample introductions, noting how the second introduction is more effective because it does not use phrases like "the topic of this essay" and expresses the writer's position confidently rather than timidly. Frequently asked questions about introductions are also addressed.
How to give a good 10min presentation Jodie Martin
This document provides tips for giving a good 10 minute presentation. It recommends choosing a theme, limiting the presentation to 10 slides with 1 slide per minute, and practicing aloud to time yourself. The key points are to educate the audience slowly and clearly while maintaining excitement through passion for the topic. Presenters should not fear questions and should conclude with confidence after covering an introduction, necessary information, and interesting details.
The document provides tips for giving introductions in formal presentations. It recommends using the WISE OWL method: Welcome the audience, Introduce yourself, Say what you'll talk about, Explain why the topic is useful, Outline the structure, discuss What materials you'll use, and Let the audience know when they can ask questions. It offers sample language for each part of the introduction and emphasizes getting the audience's attention at the start through rhetorical questions, stories, or interesting facts.
The document provides an overview of math topics that readers should already know, including properties of exponents, factoring polynomials, operations on rational expressions, operations on radicals, and linear functions. It lists specific skills under each topic, such as multiplying and dividing powers for exponents or factoring out a monomial. The document concludes by assigning practice problems from different books to reinforce these math concepts.
The document discusses the history and concepts of radicals. It explains that Pythagoras and his followers believed that natural numbers and proportions between natural numbers governed the universe. However, the Pythagorean theorem disproved this by showing the existence of irrational numbers like the square root of 2. The key points are:
- Pythagoras' philosophical theory was disproved by the existence of irrational numbers like the square root of 2 from the Pythagorean theorem.
- The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
- Radicals can be used to express solutions to equations and powers with fractional
This document provides information on radicals and root expressions. It defines the key parts of a radical expression, including the index and radicand. It explains that roots and radicals are inverse operations of exponents, and how to "undo" a power or radical. Examples are given of various nth roots and their relationships to exponents. The document outlines the rules for principal roots, noting even roots have two solutions while odd roots only have one. It cautions the reader to check the index when evaluating roots. Finally, it discusses how to simplify nth roots involving variables by dividing the exponent by the index.
The document provides an overview of basic algebra concepts including:
- Natural numbers are used for counting while whole numbers include zero. Rational numbers are numbers that can be written as fractions.
- Exponents are used to represent repeated multiplication. The order of operations is PEMDAS.
- Properties of real numbers like commutativity and inverses are used to simplify expressions. Radicals represent roots and have rules like the product rule.
This document provides an overview of key concepts in algebra, including:
- Natural numbers are used for counting, whole numbers include zero. Rational numbers are numbers that can be written as fractions.
- Exponents are used to represent repeated multiplication. The order of operations is PEMDAS.
- Properties of real numbers like commutativity and inverses apply to algebraic expressions and formulas.
- Rules of exponents allow simplifying expressions with exponents, like the product and power rules. Scientific notation uses exponents to write very large and small numbers.
This document provides an introduction to foundational algebra concepts including real numbers, integers, opposites, rational and irrational numbers, variables, absolute value, and graphing numbers on a number line. It includes vocabulary definitions and examples of evaluating expressions, comparing numbers, graphing sets of numbers, and solving absolute value expressions. The homework assigned is to complete odd problems 1-43 on page 54.
The document provides examples for dividing fractions by using the reciprocal method. It first defines a reciprocal as the "flip" of a fraction. It then shows two examples of dividing fractions step-by-step: 1) 3/7 ÷ 1/2 and 2) 6/1 ÷ 3/4. For each example, it shows finding the reciprocal of the divisor, multiplying instead of dividing, multiplying the numerators and denominators, and simplifying the final answer. The goal is to explain how to divide fractions by using reciprocals.
The document provides examples for dividing fractions by using the reciprocal method. It first defines a reciprocal as the "flip" of a fraction. It then shows two examples of dividing fractions step-by-step: 1) find the reciprocal of the divisor, 2) multiply instead of divide, and 3) multiply the numerators and denominators. This allows dividing fractions to become multiplying fractions.
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dug84356_ch09a.qxd 9/14/10 2:11 PM Page 557
’
9.1
9.2
9.3
9.4
9.5
9.6
Radicals
Rational Exponents
Adding, Subtracting, and
Multiplying Radicals
Quotients, Powers,
and Rationalizing
Denominators
Solving Equations with
Radicals and Exponents
Complex Numbers
9
Radicals and Rational
Exponents
Just how cold is it in Fargo, North Dakota, in winter? According to local meteorol
ogists, the mercury hit a low of –33°F on January 18, 1994. But air temperature
alone is not always a reliable indicator of how cold you feel. On the same date,
the average wind velocity was 13.8 miles per hour. This dramatically affected how
cold people felt when they stepped outside. High winds along with cold temper
atures make exposed skin feel colder because the wind significantly speeds up
the loss of body heat. Meteorologists use the terms “wind chill factor,”“wind chill
index,” and “wind chill temperature” to take into account both air temperature
and wind velocity.
Through experimentation in Ant
arctica, Paul A. Siple developed a
formula in the 1940s that measures the
wind chill from the velocity of the wind
and the air temperature. His complex
formula involving the square root of
the velocity of the wind is still used
today to calculate wind chill temper
atures. Siple’s formula is unlike most
scientific formulas in that it is not
based on theory. Siple experimented
with various formulas involving wind
velocity and temperature until he
found a formula that seemed to predict
how cold the air felt.
W
in
d
ch
ill
te
m
pe
ra
tu
re
(
F
)
fo
r
25
F
a
ir
te
m
pe
ra
tu
re
25
20
15
10
5
0
5
10
15
Wind velocity (mph)
5 10 15 20 25 30
Siple s formula is stated
and used in Exercises 111
and 112 of Section 9.1.
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 558
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→
→
558 Chapter 9 Radicals and Rational Exponents 9-2
9.1 Radicals
In Section 4.1, you learned the basic facts about powers. In this section, you will
study roots and see how powers and roots are related.
In This Section
U1V Roots
U2V Roots and Variables
U3V Product Rule for Radicals
U4V Quotient Rule for Radicals U1V Roots
U5V Domain of a Radical We use the idea of roots to reverse powers. Because 32 = 9 and (-3)2 = 9, both 3 andExpression or Function
-3 are square roots of 9. Because 24 = 16 and (-2)4 = 16, both 2 and -2 are fourth
roots of 16. Because 23 = 8 and (-2)3 = -8, there is only one real cube root of 8 and
only one real cube root of -8. The cube root of 8 is 2 and the cube root of -8 is -2.
nth Roots
If a = bn for a positive integer n, then b is an nth root of a. If a = b2, then b is a
square root of a. If a = b3, then b is the cube root of a.
If n is a positive even integer and a is positive, then there are two real nth roots of
a. We call these roots even roots. The positive even root of a positive number is called
the prin ...
The document discusses two geometrical theorems about parabolas:
1. It proves that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
2. It proves that the angle of incidence of a line parallel to the axis of a parabola is equal to the angle of reflection. This establishes that a line and its reflection are equally inclined to the normal and tangent of the parabola.
1) The document discusses evaluating nth roots of numbers and expressions, including square roots, cube roots, and nth roots.
2) It provides definitions and examples of even and odd roots of positive and negative numbers. Even roots of negative numbers are undefined, while odd roots of negative numbers are negative.
3) Methods are described for evaluating nth roots of monomial expressions by splitting them into factors and taking the nth root of each factor.
The document discusses nth roots and rational exponents. It defines an nth root as the value that when raised to the nth power equals a given number. nth roots can be written using rational exponents, where the denominator is the index of the radical. If n is odd, there is one real nth root, and if n is even and the number is positive, there are two real nth roots. Examples are given of finding and evaluating nth roots and solving equations involving nth roots.
Exponents represent repeated multiplication of a base number. An exponent tells how many times the base number is used as a factor. For example, in the expression 53, 5 is the base and 3 is the exponent, meaning 5 x 5 x 5 or 125. Any number to the 0 power equals 1, and any number to the 1st power equals itself. It is a common mistake to think the exponent indicates the number of factors instead of the number of times the base is used as a factor.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
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A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
1. E xam 4 M aterial
Radicals, Rational E xponents & E quations
2. Square Roots
A square root of a real number “ a” is a real number that
multiplies by itself to give “ a”
3
What is a square root of 9? 3
What is another square root of 9?
What is the square root of -4 ?
Square root of – 4 does not exist in the real number system
Why is it that square roots of negative numbers do not exist in
the real number system?
N real number multiplied by itself can give a negative answer
o
Every positive real number “ a” has two square roots that have
equal absolute values, but opposite signs 16
T two square roots of 16 are:
he
16 and simplified : 4 and 4
5 and 5
T two square roots of 5 are:
he
(Positive Square Root : PRINCIPLE ROOT)
3. Even Roots (2,4,6,…)
The even “ n th ” root of a real number “ a” is a real number that
multiplies by itself “ n” times to give “ a”
Even roots of negative numbers do not exist in the real number
system, because no real number multiplied by itself an even
number of times can give a exist
4
16 does not negative number
Every positive real number “ a” has two even roots that have
equal absolute values, but opposite signs
T fourth roots of 16:
he 4
16 and 4 16 simplified : 2 and 2
T fourth roots of 7:
he 4 4
7 and - 7
(Positive Even Root : PRINCIPLE ROOT)
4. Radical Expressions
On the previous slides we have used symbols of the form:
n
a
This is called a radical expression and the parts of the
expression are named:
Index:
Radical S ign :
n
Radicand:
E xample:
a
5
8 Index : 5 Radicand : 8
5. Cube Roots
The cube root of a real number “ a” is a real number that
multiplies by itself 3 times to give “ a”
Every real number “ a” has exactly one cube root that is
positive when “ a” is positive, and negative when “ a” is
negative
Only cube root of – 8:
Only cube root of 6:
3
8 2
3
6
No such thing as a principle cube root!
6. Odd Roots (3,5,7,…)
The odd n th root of a real number “ a” is a real number that
multiplies by itself “ n” times to give “ a”
Every real number “ a” has exactly one odd root that is
positive when “ a” is positive, and negative when “ a” is
negative
T only fifth root of - 32:
he
T only fifth root of -7:
he
3
32 2
5 7
7. Rational, Irrational, and Non-real Radical
Expressions
n
a
is non-real only if the radicand is negative and the index is
even
6
20 is non - real because radicand is negative and index is even
n
arepresents a rational number only if the radicand can be
written as a “ perfect n th ” power of an integer or the ratio of two
integers
32 is rational because 32 2
5
5
32 2 5
n a represents an irrational number only if it is a real number
and the radicand can not be written as “ perfect n th ” power of an
integer or the ratio of two integers
.
4
8 is irrational because 8 is not the fourth power 4
8
of an integer or the ratio of two integers
8. Homework Problems
S ection: 10.1
Page: 666
Problems: A ll: 1 – 6, Odd: 7 – 31, 39 – 57, 65 – 91
M yM athL ab Homework A ssignment 10.1 for practice
M yM athL ab Quiz 10.1 for grade
9. Exponential Expressions
an
“ a” is called the base
“ n” is called the exponent
If “ n” is a natural number then “ a n ” means that “ a” is to be multiplied by
itself “ n” times.
E xample: What is the value of 2 4 ?
(2)(2)(2)(2) = 16
A n exponent applies only to the base (what it touches)
E xample: What is the value of: - 3 4 ?
- (3)(3)(3)(3) = - 81
E xample: What is the value of: (- 3)4 ?
(- 3)(- 3)(- 3)(- 3) = 81
M eanings of exponents that are not natural numbers will be discussed in
this unit.
10. Negative Exponents: a -n
A negative exponent has the meaning: “ reciprocate the
base and make the exponent positive”
n
n 1
a
a
Examples:
2
2 1 1
3
3 9
3 3
2 3 27
3 2 8
.
11. Quotient Rule for Exponential Expressions
When exponential expressions with the same base are divided, the result is
an exponential expression with the same base and an exponent equal to the
numerator exponent minus the denominator exponent
am
a mn
an
E xamples:
54
547 53
57
x12
x12 4 x 8
. x4
12. Rational Exponents (a1/n)
and Roots 1
n
A n exponent of the form
has the meaning: “ the n th root of the base, if it exists,
and, if there are two nth roots, it means the principle
(positive) one”
1
th
a , if it exists, is the n root of a
n
1
(If there are two n th roots, a is the principle (positive) one)
n
1
( a multiplies by itself n times to give a)
n
13. Examples of
Rational Exponent of the Form: 1/n
1
100 2 10 (positive square root of 100)
1
5 5 (positive square root of 5)
2
1
3 2 (Does not exist! )
1
3 3 (negative square root of 3)
2
1
7 4 4
7 (positive fourth root of 7)
1
9 7 7
9 (seventh root of negative 9) .
1
8 6 (Does not exist! )
14. Summary Comments about Meaning
of a1/n
When n is odd:
a 1/n always exists and is either positive, negative or zero
depending on whether “ a” is positive, negative or zero
When n is even:
a 1/n never exists when “a” is negative
a 1/n always exists and is positive or zero depending on
whether “ a” is positive or zero
15. Rational Exponents of the Form: m/n
A n exponent of the form m/ has two equivalent
n
meanings:
(1) a m/n means find the n th root of “ a” , then raise it to
the power of “ m”
(assuming that the n th root of “ a” exists)
(2) a m/n means raise “ a” to the power of “ m” then
take the n th root of a m
(assuming that the n th root of “ a m” exists)
16. Example of Rational Exponent of the Form:
m/n
82/3
by definition number 1 this means find the cube root of 8,
then square it:
82/3 = 4
(cube root of 8 is 2, and 2 squared is 4)
by definition number 2 this means raise 8 to the power of 2
and then cube root that answer:
82/3 = 4
(8 squared is 64, and the cube root of 64 is 4)
17. Definitions and Rules for Exponents
A ll the rules learned for natural number exponents continue to
be true for both positive and negative rational exponents:
Product Rule: a ma n = a m+n 4 2 6
37 37 37
2
Quotient Rule: a m/ n = a m-n
a
3 7
2
4
3 7
Negative Exponents: a -n = (1/ n
a) 3 7
4
4
1 7
3 7
3
.
18. Definitions and Rules for Exponents
4
2
7 8
3 3
7 49
Power Rules: (a ) = a
m n mn
2 2
2
(ab) = a b
m m m
3x 7 3 x
7 7
2
2
(a/ m = a m /b m
b) 3 3 7
7
2
4 47
0
Zero Exponent: a 0 = 1 (a not zero) 3
1
. 4
19. “Slide Rule” for Exponential Expressions
When both the numerator and denominator of a
fraction are factored then any factor may slide from
the top to bottom, or vice versa, by changing the sign
on the exponent
Example: Use rule to slide all factors to other part of
the fraction:
a mb n cr d s
r s
m n
c d a b
This rule applies to all types of exponents
Often used to make all exponents positive
20. Simplifying Products and Quotients Having
Factors with
Rational Exponents
All factors containing a common base can be combined using
rules of exponents in such a way that all exponents are
positive:
Use rules of exponents to get rid of parentheses
S implify top and bottom separately by using product rules
Use slide rule to move all factors containing a common base to
the same part of the fraction
If any exponents are negative make a final application of the
slide rule
21. Simplify the Expression:
1
8 y y
3
2
8
8y
3 16
3 1 7
1
2 1 y 12 39
2 y y 4 6
2
y 12
8 y y 23 21 8
3 1 7 8
2 1 y y 4 6 y y
12 3
2 6
8y y 3 3 16
9 2 7 32
2 1 y y
12 12 y y
12 12
22. Applying Rules of Exponents
in Multiplying and Factoring
Multiply: 1 1
1
1 1
1
1 1
1
x 2 2 x 2 x 2 x 2 x 2 x 2 x 2 2 x 2 2 x 2
1 1 1 1 1 1 1 1
x 2 2
x 2 2
2x 2x
2 2 x 0 x 1 2 x 2 x 2 2
1 1
1 x 1 2 x 2 x
2 2
Factor out the indicated factor:
3 1
3
5x 4
x ;x 4 4
3 3 4
3
5 x
x 4
__ __ 4 4
x 5 x
x 4
23. Radical Notation
Roots of real numbers may be indicated by means of
either rational exponent notation or radical notation:
n
a is called a RADICAL (expressio n)
is called a RADICAL SIGN
n is called the INDEX
a is called the RADICAND
24. Notes About Radical Notation
If no index is shown it is assumed to be 2
When index is 2, the radical is called a “ square root”
When index is 3, the radical is called a “ cube root”
When index is n, the radical is called an “ nth root”
In the real number system, we can only find even roots of
non-negative radicands. There are always two roots when the
index is even, but a radical with an even index always means
the positive (principle) root
We can always find an odd root of any real number and the
result is positive or negative depending on whether the radicand
is positive or negative
25. Converting Between Radical and Rational
Exponent Notation
A n exponential expression with exponent of the form “ m/n”
can be converted to radical notation with index of “ n” , and
vice versa, by either of the following formulas:
m 2
a a
n n m
83 3
82 3 64 4
1.
a 8
m 2
2 4
m 2 2
2. a
n n
8
3 3
These definitions assume that the nth root of “ a” exists
27. n n
x
.
If “ n” is even, then this notation means principle
(positive) root:
n
x x
n
(absolute value needed to insure positive answer)
If “ n” is odd, then:
x x
n n
If we assume that “ x” is positive (which we often do) then
we can say that:
.
n
x xn
28. Homework Problems
S ection: 10.2
Page: 675
Problems: A ll: 1 – 10, Odd: 11 – 47, 51 – 97
M yM athL ab Homework A ssignment 10.2 for practice
M yM athL ab Quiz 10.2 for grade
29. Product Rule for Radicals
When two radicals are multiplied that have the same index
they may be combined as a single radical having that
index and radicand equal to the product of the two
radicands:
This rule works both directions:
n
a b ab
n n 4
3 5 3 5 15
4 4 4
n
ab a bn n 3
16 8 2 2 2
3 3 3
30. Quotient Rule for Radicals
When two radicals are divided that have the same index
they may be combined as a single radical having that
index and radicand equal to the quotient of the two
radicands 4
n
a n a 5 5
4
n
b b 4
7 7
This rule works both directions:
a na 5 35 3
5
n n 3 3
. b b 8 8 2
31. Root of a Root Rule for Radicals
When you take the m th root of the n th root of a radicand
“ a” , it is the same as taking a single root of “ a” using an
index of “ mn”
m n
a mn
a
4 3
6 12
6
.
32. NO Similar Rules for Sum and Difference of
Radicals
n
a n b n ab .
3
27 8 35
3 3
3 2 35
3
n
a n b n a b
3
27 3 8 3 19
3 2 19 3
33. Simplifying Radicals
A radical must be simplified if any of the following
conditions exist:
2. S ome factor of the radicand has an exponent that
is bigger than or equal to the index
3. There is a radical in a denominator (denominator
needs to be “ rationalized” )
4. The radicand is a fraction
5. All of the factors of the radicand have exponents
that share a common factor with the index
34. Simplifying when Radicand has
3 4
2
Exponent Too Big
1. Use the product rule to write the single radical as a
product of two radicals where the first radicand
contains all factors whose exponents match the index
and the second radicand contains all other factors
2. S implify the first radical
3 33
2 2
3
2 2
35. Problem?
Example
2 5
3
24 x y Is there another exponent t hat is too big?
3 2 5
3
2 3x y Write this as a product of two radicals :
3 33 2 2
3
2 y 3x y Simplify the first radical :
2 2
2 y 3x y
3
36. Simplifying when a Denominator Contains
a Single Radical of Index “n”
1. S implify the top and bottom separately to get rid of exponents
under the radical that are too big
2. M ultiply the whole fraction by a special kind of “ 1” where 1
is in the form of: n
m
n
m
and m is the product of all the factors required to
make every exponent in the radicand be equal to quot;nquot;
7. S implify to eliminate the radical in the denominator
37. Example 3
3 3 3
3 6 2 3 6 55 2 3
5
4x y 5
2 x y 5
y 2 x y y 5 22 x 3 y
3 5
23 x 2 y 4 35 23 x 2 y 4 35 23 x 2 y 4
y5 22 x3 y 5
23 x 2 y 4 y 5 25 x 5 y 5 2 xy 2
2 4
3 8x y
5
2
2 xy
38. Simplifying when Radicand is a
Fraction
1. Use the quotient rule to write the single radical as a
quotient of two radicals
2. Use the rules already learned for simplifying when
there is a radical in a denominator
40. Simplifying when All Exponents in
Radicand Share a Common Factor with
Index
1. Divide out the common factor from the index and
all exponents
4 6 8 2
6
23 x y
All exponents in radicand and index share what factor? 2
Dividing all exponents in and index by 2 gives :
3 2 3 4
23 x y 3 x 3 3 33 2
2 xy 3 x 3 4 xy
Problem?
41. Simplifying Expressions Involving Products
and/or Quotients of Radicals with the Same
Index
Use the product and quotient rules to combine everything
under a single radical
Simplify the single radical by procedures previously
discussed
42. Example
4
ab ab 34
a 2b 4 b 4
b 4
b 4
a3
4 3 3 4 4 4
4
a 3b 3 ab a a a 4
a3
4 3
4
ab 3
ab
4
a4
a
43. Right Triangle
A “ right triangle” is a triangle that has a 90 0 angle (where
two sides intersect perpendicularly)
b c hypotenuse
90 0
The side opposite the right angle is called the
a
“ hypotenuse” and is traditionally identified as side “ c”
The other two sides are called “ legs” and are traditionally
labeled “ a” and “ b”
44. Pythagorean Theorem
In a right triangle, the square of the hypotenuse is always
equal to the sum of the squares of the legs:
c a b
2 2 2
c
b
90 0
a
45. Pythagorean Theorem Example
It is a known fact that a triangle having shorter sides of
lengths 3 and 4, and a longer side of length 5, is a right
triangle with hypotenuse 5.
5
Note that Pythagorean Theorem 3 true:
is 90 0
4
c a b
2 2 2
5 4 3
2 2 2
25 16 9
46. Using the Pythagorean Theorem
We can use the Pythagorean Theorem to find the third
side of a right triangle, when the other two sides are
known, by finding, or estimating, the square root of a
number
47. Using the Pythagorean Theorem
Given two sides of a right triangle with one side unknown:
Plug two known values and one unknown value into
Pythagorean Theorem
Use addition or subtraction to isolate the “ variable squared”
S quare root both sides to find the desired answer
48. Example
Given a right triangle with
a 7 and c 25
find the other
side.
c a b
2 2 2
25 7 b
2 2 2
625 49 b 2
625 49 49 49 b 2
576 b 2
24 576 b
49. Homework Problems
S ection: 10.3
Page: 685
Problems: Odd: 7 – 19, 23 – 57, 61 – 107
M yM athL ab Homework A ssignment 10.3 for practice
M yM athL ab Quiz 10.3 for grade
50. Adding and Subtracting Radicals
Addition and subtraction of radicals can always be indicated,
but can be simplified into a single radical only when the
radicals are “like radicals”
“Like Radicals” are radicals that have exactly the same index
and radicand, but may have different coefficients
4 4 3
Which are like radicals? 3 5 , 4 5 , - 2 5 and 3 5
When “like radicals” are added or subtracted, the result is a
“like radical” with coefficient equal to the sum or difference of
the coefficients
3 5 2 5 54 5
4 4
-2 5 3 5
4 3 Okay as is - can' t combine unlike radicals
51. Note Concerning Adding and Subtracting
Radicals
When addition or subtraction of radicals is indicated you
must first simplify all radicals because some radicals that
do not appear to be like radicals become like radicals
when simplified
53. Homework Problems
S ection: 10.4
Page: 691
Problems: Odd: 5 – 57
M yM athL ab Homework A ssignment 10.4 for practice
M yM athL ab Quiz 10.4 for grade
54. Simplifying when there is a Single Radical
Term in a Denominator
1. Simplify the radical in the denominator
2. If the denominator still contains a radical, multiply
the fraction by “ 1” where “ 1” is in the form of a
“special radical” over itself
3. The “ special radical” is one that contains the factors
necessary to make the denominator radical factors
have exponents equal to index
4. Simplify radical in denominator to eliminate it
55. Example
2 3
Simplify denominato r :
3
9x
1
3
2
3 2
3 2
3 x
Multiply by special quot;1quot;:
6x
3
2 3
3x 2
Use product rule :
3x
3 2 3 2
3 x 3x
3
2 3x 2 Simplify denominato r :
3 3 3
3 x
56. Simplifying to Get Rid of a Binomial Denominator
that Contains One or Two Square Root Radicals
1. Simplify the radical(s) in the denominator
2. If the denominator still contains a radical, multiply
the fraction by “ 1” where “ 1” is in the form of a
“special binomial radical” over itself
3. The “ special binomial radical” is the conjugate of
the denominator (same terms – opposite sign)
4. Complete multiplication (the denominator will
contain no radical)
57. Example Radical in denominator doesn' t need simplifying
5
3 2 Multiply fraction by special one :
5 3 2 Distribute on top :
3 2 3 2 FOIL on bottom :
15 10
9 4 Simplify bottom :
15 10
3 2
15 10
58. Homework Problems
S ection: 10.5
Page: 700
Problems: Odd: 7 – 105
M yM athL ab Homework A ssignment 10.5 for practice
M yM athL ab Quiz 10.5 for grade
59. Radical Equations
A n equation is called a radical equation if it contains a
variable in a radicand
E xamples:
x x3 5
x x 5 1
3
x 4 3 2x 0
60. Solving Radical Equations
1. Isolate ONE radical on one side of the equal sign
2. Raise both sides of equation to power necessary to
eliminate the isolated radical
3. Solve the resulting equation to find “ apparent
solutions”
4. Apparent solutions will be actual solutions if both
sides of equation were raised to an odd power,
BUT if both sides of equation were raised to an
even power, apparent solutions MUST be
checked to see if they are actual solutions
61. Why Check When Both Sides are Raised to
an Even Power?
Raising both sides of an equation to a power does not always result in
equivalent equations
If both sides of equation are raised to an odd power, then resulting
equations are equivalent
If both sides of equation are raised to an even power, then resulting
equations are not equivalent (“ extraneous solutions” may be introduced)
Raising both sides to an even power, may make a false statement true:
2 2 , however : - 2 2 , - 2 2 , etc.
2 2 4 4
Raising both sides to an odd power never makes a false statement true:
2 2 , and : - 2 2 , - 2 2 , etc. .
3 3 5 5
62. Example of Solving
Radical Equation Check x 4
x x3 5
4 43 5?
x5 x3
4 1 5?
x 5 2
x3 2
35
x 4 is NOT a solution
x 10 x 25 x 3
2
Check x 7
x 2 11x 28 0 7 73 5?
x 4 x 7 0 7 4 5?
x 4 0 OR x 7 0 55
x 4 OR x 7 x7 IS a solution
63. Example of Solving
Radical Equation
x x 5 1 Check x 4
x 5 1 x 4 4 5 1?
2
x 5 1 x 2
4 9 1?
x 5 1 2 x x 2 3 1?
4 2 x 5 1
x 4 is NOT a solution
2 x
2 x
2 2
Equation has No Solution!
4x
64. Example of Solving
Radical Equation
3
x 4 3 2x 0
3
x 4 2x
3
3
x4
3
3
2x 3
x 4 2x
4x
(No need to check)
65. Homework Problems
S ection: 10.6
Page: 709
Problems: Odd: 7 – 57
M yM athL ab Homework A ssignment 10.6 for practice
M yM athL ab Quiz 10.6 for grade