7.1 nth roots and rational exponents
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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.
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Chapter 6.1
This document contains examples of solving equations involving exponents and roots. It discusses the number of solutions when solving nth roots depending on if n is even or odd and if the radicand is positive, negative or zero. It also contains examples of solving equations with rational exponents and integer exponents.
7.7 Solving Radical Equations
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
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This document defines exponents and radicals. It discusses exponential notation, zero and negative exponents, and the laws of exponents. It also covers scientific notation, nth roots, rational exponents, and rationalizing the denominator. The objectives are to define integer exponents and exponential notation, zero and negative exponents, identify laws of exponents, write numbers using scientific notation, and define nth roots and rational exponents.
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7.7 Solving Radical Equations
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
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This document defines exponents and radicals. It discusses exponential notation, zero and negative exponents, and the laws of exponents. It also covers scientific notation, nth roots, rational exponents, and rationalizing the denominator. The objectives are to define integer exponents and exponential notation, zero and negative exponents, identify laws of exponents, write numbers using scientific notation, and define nth roots and rational exponents.
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This document provides information about polygons and quadrilaterals, including:
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joint variation
z = kxy
z = -12
z = kxy
z = -84
z = kxy
z = -21
4.
a. Combined means together as a whole.
b. Combined variation is when a quantity varies jointly with respect to the product of two or more variables.
c. The mathematical statement that represents combined variation is a = k(bc) where a varies jointly as b and c multiplied together.
Joint and Combined Variation (Mathematics 9)
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2) f(n) = 2n - 1 for the sequence 1, 3, 5, 7,...
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The document provides instructions and examples for simplifying expressions with rational exponents. It asks students to write expressions using rational exponents, identify necessary skills for simplifying such expressions, fill in missing parts of solutions, and simplify given expressions showing the boxed final answer. Students are directed to work in groups to present and prepare for examples, and to simplify expressions using positive exponents to decode clues about a garden location.
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The document defines direct and inverse variation and provides examples of translating statements describing direct and inverse variation into mathematical equations. It also provides two examples of using the equations to solve for unknown variables given specific values of other variables. Specifically, it defines direct variation as being proportional and inverse variation as being inversely proportional. It translates statements about direct and inverse variation into equations using a constant of variation k. The examples show setting the equations equal to known values and solving for the constant k and then using k to solve for the unknown variable.
Synthetic division
This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15
Roots and radical expressions
This document discusses operations involving radicals such as square roots. Some key points covered include:
1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical.
2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same.
3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator.
4) Adding and subtracting radicals by combining like terms where only the numbers outside the radicals are added/subtracted and numbers inside are kept the same.
Module 14 lesson 4 solving radical equations
This document provides instructions on how to solve radical equations. It begins by defining a radical equation and radicand. It then explains the main steps to solve radical equations: isolate the radical to one side, raise both sides to the same power, and simplify. Several examples are worked through to demonstrate this process. The document emphasizes checking solutions, as raising both sides to an even power can result in extraneous solutions. TI instructions and additional practice problems are also included.
Extracting the roots
The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
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2. The common difference is found by subtracting the first term from the second term. This yields the constant value that is added to each term to generate the next term.
3. Formulas can be used to find a specific term in an arithmetic sequence or the sum of terms, given the first term, common difference, and number of terms.
Quadratic inequality
This document discusses solving quadratic inequalities. It provides examples of single-variable quadratic inequalities and explains how to find the solution set by first setting the inequality equal to an equation, solving for the roots, and then testing values within the intervals formed by the roots. The document also introduces quadratic inequalities with two variables and how to represent them. It defines a quadratic inequality and explains the process of solving them by relating it back to solving a quadratic equation.
Factoring perfect-square-trinomials
To factor perfect square trinomials:
1. Get the square root of the first and last terms.
2. List down the square root as the sum or difference of two terms.
3. Factor the expression into the form (term 1 ± term 2)2.
Solving quadratics by completing the square
The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.
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The document discusses radicals and their properties. It defines radicals as irrational numbers expressed in roots, and another way to express numbers with fractional exponents. Radicals are expressed as an, where n is the index and a is the radicand. The document asks questions about what affects the quadratic term and constant term of a quadratic function. It also relates the multiplier of the quadratic term to the width of the parabola.
U2 05 Radicacion
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
Chapterdug84356_ch09a.qxd 91410 211.docx
C
h
a
p
t
e
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’
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 ...
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- Parabolas can be represented using various equation forms including vertex form, standard form, and general form.
- Methods for graphing parabolas by identifying features like the vertex, axis of symmetry, x-intercepts, focus, and directrix.
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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.
Exponential equations
An exponential equation is one where a variable appears in the exponent. To solve exponential equations without logarithms, both sides must have comparable exponential expressions with the same base, so the powers can be set equal. If both sides can be expressed as powers of the same base b, the property b^x = b^y implies x = y can be used to set the exponents equal and solve for the variable. This demonstrates how exponential equations are solved by setting the powers equal when the bases are the same. For equations that cannot be expressed with a common base, logarithms are used.
joint variation
z = kxy
z = -12
z = kxy
z = -84
z = kxy
z = -21
4.
a. Combined means together as a whole.
b. Combined variation is when a quantity varies jointly with respect to the product of two or more variables.
c. The mathematical statement that represents combined variation is a = k(bc) where a varies jointly as b and c multiplied together.
Joint and Combined Variation (Mathematics 9)
You already know relationships where one variable varies directly or inversely with another variable.
Now you will look at relationships where one variable varies directly with two or more other variables but does not vary inversely with any other variable.
General term
The document provides examples of finding the general term of different sequences. For each sequence, it identifies the pattern between the terms and determines the formula for the nth term. The general terms provided are:
1) f(n) = 2n for the sequence 2, 4, 6, 8,...
2) f(n) = 2n - 1 for the sequence 1, 3, 5, 7,...
3) f(n) = 2n for the sequence 2, 4, 8, 16,...
4) f(n) = n^2 for the sequence 1, 4, 9, 16,...
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Graph of linear equations
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
Zero exponents, negative integral exponents, rational
The document provides instructions and examples for simplifying expressions with rational exponents. It asks students to write expressions using rational exponents, identify necessary skills for simplifying such expressions, fill in missing parts of solutions, and simplify given expressions showing the boxed final answer. Students are directed to work in groups to present and prepare for examples, and to simplify expressions using positive exponents to decode clues about a garden location.
COMBINED VARIATION.pptx
The document defines direct and inverse variation and provides examples of translating statements describing direct and inverse variation into mathematical equations. It also provides two examples of using the equations to solve for unknown variables given specific values of other variables. Specifically, it defines direct variation as being proportional and inverse variation as being inversely proportional. It translates statements about direct and inverse variation into equations using a constant of variation k. The examples show setting the equations equal to known values and solving for the constant k and then using k to solve for the unknown variable.
Synthetic division
This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15
Roots and radical expressions
This document discusses operations involving radicals such as square roots. Some key points covered include:
1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical.
2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same.
3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator.
4) Adding and subtracting radicals by combining like terms where only the numbers outside the radicals are added/subtracted and numbers inside are kept the same.
Module 14 lesson 4 solving radical equations
This document provides instructions on how to solve radical equations. It begins by defining a radical equation and radicand. It then explains the main steps to solve radical equations: isolate the radical to one side, raise both sides to the same power, and simplify. Several examples are worked through to demonstrate this process. The document emphasizes checking solutions, as raising both sides to an even power can result in extraneous solutions. TI instructions and additional practice problems are also included.
Extracting the roots
The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
Arithmetic Sequence
1. An arithmetic sequence is a sequence where each term is calculated by adding a fixed amount, called the common difference, to the previous term.
2. The common difference is found by subtracting the first term from the second term. This yields the constant value that is added to each term to generate the next term.
3. Formulas can be used to find a specific term in an arithmetic sequence or the sum of terms, given the first term, common difference, and number of terms.
Quadratic inequality
This document discusses solving quadratic inequalities. It provides examples of single-variable quadratic inequalities and explains how to find the solution set by first setting the inequality equal to an equation, solving for the roots, and then testing values within the intervals formed by the roots. The document also introduces quadratic inequalities with two variables and how to represent them. It defines a quadratic inequality and explains the process of solving them by relating it back to solving a quadratic equation.
Factoring perfect-square-trinomials
To factor perfect square trinomials:
1. Get the square root of the first and last terms.
2. List down the square root as the sum or difference of two terms.
3. Factor the expression into the form (term 1 ± term 2)2.
Solving quadratics by completing the square
The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and how to find the missing constant term to create them. The steps for solving a quadratic equation by completing the square are described: 1) move all terms to the left side, 2) find and add the "completing the square" term, 3) factor into a perfect square trinomial, 4) take the square root of each side, 5) solve the resulting equations. Additional examples demonstrate applying these steps to solve quadratic equations algebraically.
Quadratic function
The document discusses quadratic functions and their zeros. It provides examples of finding the zeros of quadratic functions by factoring, completing the square, and using the quadratic formula. It also gives examples of writing the equation of a quadratic function given its zeros or given properties like the vertex and y-intercept. Methods discussed include using the fact that the zeros are the roots of the corresponding equation, and substituting known point values into the quadratic formula.
Factor Theorem and Remainder Theorem
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Radical expressions
The document discusses radicals and their properties. It defines radicals as irrational numbers expressed in roots, and another way to express numbers with fractional exponents. Radicals are expressed as an, where n is the index and a is the radicand. The document asks questions about what affects the quadratic term and constant term of a quadratic function. It also relates the multiplier of the quadratic term to the width of the parabola.
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Zero exponents, negative integral exponents, rational
Zero exponents, negative integral exponents, rational
Similar to 7.1 nth roots and rational exponents
U2 05 Radicacion
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
Chapterdug84356_ch09a.qxd 91410 211.docx
C
h
a
p
t
e
r
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
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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 ...
My own exp nd radi
This document discusses exponents, radicals, and nth roots. It defines exponential notation, laws of exponents, properties of radicals, and how to simplify expressions with exponents and radicals. It also explains what an nth root is, how to write nth roots using rational exponents, and how to determine the number of real nth roots a number has based on whether the index is odd or even. Examples are provided to illustrate how to solve equations and simplify expressions involving exponents, radicals, and nth roots.
M14 T1
- The document discusses perfect squares, square roots, cubes, and nth roots.
- It provides examples of finding square roots and cube roots of various numbers.
- Key formulas introduced include the volume of a sphere and estimating distance to the horizon.
Roots of real numbers and radical expressions
This document defines roots and radical expressions. It defines a square root as a number that produces another number when squared, and defines an nth root as a number that produces another number when raised to the nth power. It discusses notation for radicals, simplifying radicals, principal roots, and examples of taking roots of variable expressions and using absolute value signs with radicals.
Derivation of a prime verification formula to prove the related open problems
In this document, we will develop a new formula to calculate prime numbers and use it to discuss open problems like Goldbach, Polignac and Twin prime conjectures, perfect numbers, the existence of odd harmonic divisors, ...
Note: Some people found already errors in this document. I thank them for reporting them to me. Though, I am able to solve them, I deliberately want to keep these errors in the document for the time being to discourage error seekers from reading my papers. These people look at the details while missing the bigger picture.
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6.1
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8.3 the number e
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Exponents Intro with Practice.ppt
The document discusses exponents and laws regarding operations with exponents. It defines exponents as indicating the number of times a base is multiplied by itself. The laws covered include: keeping the base and adding exponents when multiplying powers; keeping the base and subtracting exponents when dividing powers; multiplying exponents when raising a power to an exponent; applying the exponent to all factors when taking a power of a product or quotient; changing the base to its reciprocal and making the exponent positive when the exponent is negative; and any base with a zero exponent equals one. Examples are provided to demonstrate how to apply each law.
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The document discusses different types of numbers including whole numbers, natural numbers, integers, rational numbers, and real numbers. It then covers the order of operations using PEMDAS and provides examples. Finally, it discusses properties of exponents such as multiplying, dividing, and raising exponents to other exponents.
Math Project
The document discusses different types of real numbers. Natural numbers can be defined as either positive integers or non-negative integers. Whole numbers are used interchangeably to refer to non-negative integers, positive integers, or all integers. Rational numbers are numbers that can be expressed as fractions of integers, while irrational numbers like √2 or π have decimal expansions that continue forever without repeating. Almost all real numbers are irrational since the set of rational numbers is countable but the set of real numbers is uncountable.
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The document discusses different types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Natural numbers can be defined as positive integers or non-negative integers. Whole numbers are sometimes used to refer to non-negative integers, positive integers, or all integers. Rational numbers are numbers that can be expressed as fractions, while irrational numbers like √2 have decimal expansions that continue forever without repeating.
Finite mathematics
This document provides an overview of the topics that will be covered in a finite mathematics course, including residue arithmetic, elements of finite groups/rings/fields, number theory concepts like the Euclidean algorithm and Chinese Remainder Theorem, and basics of finite vector spaces and fields. The style of the course will be leisurely and discursive, focusing on mathematical thinking and discovery. While the mathematics is classical, it will be new to students. The goal is to emphasize elegance and aesthetics over utility alone.
Basics about exponents
Exponents are used to represent repeated multiplication of a number, called the base. The exponent indicates how many times the base is multiplied. Some key laws for exponents include: when multiplying the same bases, add the exponents; when dividing the same bases, subtract the exponents; when raising a power to another power, multiply the exponents. Exponents provide a shorthand way to write very large or small numbers.
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7.3 rational exponents
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The document introduces the quadratic formula as a method for solving any quadratic equation in the standard form of ax^2 + bx + c = 0. It provides the quadratic formula, and shows an example of using it to solve for x in a quadratic equation. Users are then prompted to try solving for x in a quadratic equation using the quadratic formula.
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4.2 solving for missing sides
This document provides instructions for solving for a missing side of a triangle using trigonometric functions. It tells the reader to label the sides of the triangle, set up an equation using the information given, and solve for the variable to find the missing side, making sure a calculator is in degree mode.
4.1 trig ratios
This document defines and explains the three main trigonometric ratios - sine, cosine, and tangent - which are used to solve for missing angles and sides of right triangles. It identifies the hypotenuse as the longest side, and the adjacent and opposite sides in relation to a given angle. The document provides the ratio equations and examples of calculating trig ratios for various angles in a right triangle.
R.4 solving literal equations
This document defines a literal equation as an equation with two or more different variables. It provides examples of literal equations such as A = bh, where b is solved for, and P = 2L + 2W, where W is solved for. The document demonstrates how to solve various literal equations for the indicated variable, such as solving for P in I = Prt, solving for x in A = x + y^2, and solving for x and y in two additional equations.
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This document provides instructions for solving inequalities and graphing their solutions. It notes that when multiplying or dividing an inequality by a negative number, the inequality symbol is reversed. Open circles are used for < and >, while closed circles are used for ≤ and ≥. Examples are provided of solving various inequalities algebraically and graphing the resulting solution sets on a number line.
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10 3 double and half-angle formulas
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2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
6.7 other methods for solving
This document discusses two additional methods for solving quadratic equations: the quadratic formula and graphing. The quadratic formula can be used to solve any quadratic equation in standard form (ax^2 + bx + c = 0) by inputting the a, b, and c coefficients. Graphing involves changing the equation to y= form, plotting it on a calculator to find the x-intercepts, which are the solutions. Both methods are demonstrated with examples.
10 2 sum and diff formulas for tangent
1) Formulas are derived for tangent of the sum and difference of two angles using previous trigonometric identities: tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ) and tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ).
2) These formulas are valid when the tangents of the individual angles and their sum/difference are defined, which excludes cases where the cosine of any angle is 0.
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7.1 nth roots and rational exponents
- 1. 7.1 nth Roots and Rational Exponents
- 2. What is an “nth Root?” Extends the concept of square roots. For example: ◦ A cube root of 8 is 2, since 23 = 8 ◦ A fourth root of 81 is 3, since 34 = 81 For integers n greater than 1, if bn = a then b is an nth root of a. Written where n is the index of the radical.
- 3. Rational Exponents nth roots can be written using rational exponents. For example: In general, for any integer n greater than 1.
- 4. Real nth Roots If n is odd: ◦ a has one real nth root If n is even: ◦ And a > 0, a has two real nth roots ◦ And a = 0, a has one nth root, 0 ◦ And a < 0, a has no real nth roots
- 5. Finding nth Roots Find the indicated real nth root(s) of a. Example: n = 3, a = -125 n is odd, so there is one real cube root: (- 5)3 = -125 We can write
- 6. Example: n = 4, a = 16 n is even, and a > 0, so 16 has two real 4th roots: 24 = 16 and (-2)4 = 16 We can write
- 7. Your Turn! Find the indicated real nth root(s) of a. n = 4, a = 625 n = 3, a = -27
- 8. Rational Exponents Not always of the form 1/n. The denominator of the exponent is the index of the radical.
- 9. Evaluating with Rational Exponents Evaluate: 93/2 32-2/5
- 10. Your Turn! Evaluate: 493/2 16-3/4
- 11. Approximating with a Calculator Rewrite in rational exponent notation, then use calculator. Example: Use ( ) around the exponent! Your Turn! Approximate
- 12. Solving Equations with nth Roots Get the power alone, then take nth roots of both sides. Examples: 2x4 = 162 (x – 2)3 = 10
- 13. Your Turn! Solve each equation. 5x4 = 80 (x – 1)3 = 32