Chapter4.1
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The document provides examples and explanations of exponents and exponential notation. It includes examples of writing expressions in exponential form, simplifying expressions with exponents, using the order of operations with exponents, and an application involving finding the number of diagonals in a polygon using an exponential expression.
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Showing the associative property of addition
Ben harvested 73 eggplant and 94 pieces of okra, for a total of 73 + 94 = 167 pieces of vegetables harvested. The associative property of addition states that changing the grouping of the addends does not change the sum, as shown in examples such as (4 + 3) + 5 = 4 + (3 + 5).
Math 7 lesson 11 properties of real numbers
At the end of the lesson, the learner should be able to:
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Here are the properties for each expression:
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2) Commutative Property of Multiplication
3) Associative Property of Addition
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5) Multiplicative Property of Zero
6) Identity Property of Multiplication
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The document provides examples for multiplying, dividing, and raising monomials to a power. It includes step-by-step work for multiplying terms like (3a2)(4a5), dividing terms like 15m5/3m2, and raising terms to a power like (2a2b6)4. The examples are followed by a short quiz assessing these skills.
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Showing the associative property of addition
Ben harvested 73 eggplant and 94 pieces of okra, for a total of 73 + 94 = 167 pieces of vegetables harvested. The associative property of addition states that changing the grouping of the addends does not change the sum, as shown in examples such as (4 + 3) + 5 = 4 + (3 + 5).
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At the end of the lesson, the learner should be able to:
recall the different properties of real numbers
write equivalent statements involving variables using the properties of real numbers
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The document discusses the commutative and associative properties of real numbers. The commutative property states that the order of numbers does not matter in addition and multiplication, but it does matter in subtraction and division. The associative property states that the grouping of numbers does not matter in addition and multiplication, but it does matter in subtraction and division. Both properties only apply to addition and multiplication, not subtraction and division.
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The document discusses three mathematical properties: the associative property, which allows changing the grouping of operations without changing the result; the commutative property, which allows changing the order of operations without changing the result; and the distributive property, which distributes multiplication over addition or subtraction. It provides examples of how to use each property to solve problems involving areas, discounts, and total costs.
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This document contains algebraic proofs with missing steps and reasons. It asks to name the property that justifies each statement in several proofs and to write two-column proofs for two additional proofs. The proofs involve solving equations for a variable by using properties like adding/subtracting the same quantity to both sides, multiplying/dividing both sides by the same non-zero quantity, and isolating the variable.
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Here are the properties for each expression:
1) Commutative Property of Addition
2) Commutative Property of Multiplication
3) Associative Property of Addition
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5) Multiplicative Property of Zero
6) Identity Property of Multiplication
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- The opposite property which involves changing all addition signs to subtraction or multiplying the entire expression by -1.
- The distributive property which allows multiplying a number by the terms inside parentheses.
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Chapter1.1
The document provides examples and explanations for evaluating algebraic expressions. It introduces key vocabulary like expression, variable, numerical expression, algebraic expression, and evaluate. Examples are provided for evaluating expressions with one variable, two variables, and applications involving converting between Celsius and Fahrenheit temperatures. Practice problems are included at the end to evaluate expressions for given variable values.
Ch. 4, Sec 2: Exponents
This document provides an overview of exponents and exponential notation:
- Exponents show repeated multiplication, with the base being the factor and the exponent showing how many times the base is multiplied by itself.
- Exponential notation compactly writes repeated multiplication, such as 2^6 representing 2 x 2 x 2 x 2 x 2 x 2.
- Negative integers raised to an even power are positive, while raised to an odd power are negative.
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- Order of operations must be followed when evaluating expressions with exponents.
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The document discusses dividing integers and their rules:
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- Examples are provided to demonstrate applying the rules.
- The mean, or average, of a data set is calculated by summing the values and dividing by the number of values.
- Order of operations must be followed when evaluating expressions.
Msm1 fl ch01_03
The document provides a lesson on evaluating numerical expressions using the order of operations. It includes examples of simplifying expressions with addition, subtraction, multiplication, division and exponents. It also has examples of word problems involving amounts of money spent on items. The lesson concludes with a quiz to assess understanding of simplifying expressions and applying the order of operations.
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1. The document provides information about precalculus chapter 2, which covers exponents and radicals, polynomials, factoring, and complex numbers.
2. Key topics include scientific notation, properties of exponents and radicals, adding/subtracting/multiplying polynomials, factoring polynomials, and performing operations with complex numbers.
3. Examples are provided for simplifying expressions with exponents, factoring trinomials and polynomials, using long division and synthetic division to divide polynomials, and solving quadratic equations with complex number solutions.
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- 1. Warm Up California Standards Lesson Presentation Preview
- 2. Warm Up Find the product. 625 1. 5 • 5 • 5 • 5 2. 3 • 3 • 3 3. (–7) • (–7) • (–7) 4. 9 • 9 27 – 343 81
- 3. AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Also covered: AF1.2 California Standards
- 4. Vocabulary exponential form exponent base power
- 5. If a number is in exponential form , the exponent represents how many times the base is to be used as a factor. A number produced by raising a base to an exponent is called a power . 27 and 3 3 are equivalent. 7 Exponent Base 2
- 6. Identify how many times 4 is a factor. Write in exponential form. Additional Example 1: Writing Exponents A. 4 • 4 • 4 • 4 Identify how many times –6 is a factor. (–6) • (–6) • (–6) = (–6) 3 B. (–6) • (–6) • (–6) 4 • 4 • 4 • 4 = 4 4 Read (–6) 3 as “–6 to the 3rd power" or "–6 cubed”. Reading Math
- 7. Identify how many times 5 and d are each used as a factor. Additional Example 1: Writing Exponents C. 5 • 5 • d • d • d • d Write in exponential form. 5 • 5 • d • d • d • d = 5 2 d 4
- 8. Identify how many times x is a factor. Write in exponential form. Check It Out! Example 1 A. x • x • x • x • x Identify how many times d is a factor. B. d • d • d x • x • x • x • x = x 5 d • d • d = d 3
- 9. Identify how many times 7 and b are each used as a factor. 7 • 7 • b • b = 7 2 b 2 Check It Out! Example 1 C. 7 • 7 • b • b Write in exponential form.
- 10. = 243 3 5 = 3 • 3 • 3 • 3 • 3 Find the product. Find the product. B. Simplify. Additional Example 2: Simplifying Powers A. 3 5 = 1 27
- 11. = 256 – 2 8 = –(2 • 2 • 2 • 2 • 2 • 2 • 2 • 2) = –256 Simplify. Additional Example 2: Simplifying Powers Find the product. Find the product. Then make the answer negative. D. – 2 8 = (–4) • (–4) • (–4) • (–4) (–4) 4 C. (–4) 4
- 12. The expression (–4) 4 is not the same as the expression –4 4 . Think of –4 4 as –1 ● 4 4 . By the order of operations, you must evaluate the exponent before multiplying by –1 . Caution!
- 13. = 2401 7 4 = 7 • 7 • 7 • 7 Find the product. Simplify. Check It Out! Example 2 Find the product. A. 7 4 B. = 1 8
- 14. = 25 – 9 4 = –(9 • 9 • 9 • 9) = –6,561 Evaluate. Find the product. Find the product. Then make the answer negative. Check It Out! Example 2 D. – 9 4 = (–5) • ( – 5) (–5) 2 C. (–5) 2
- 15. Additional Example 3: Using the Order of Operations 4(7) + 16 Substitute 4 for x, 2 for y, and 3 for z. Simplify the powers. Subtract inside the parentheses. Multiply from left to right. 4(2 4 – 3 2 ) + 4 2 4(16 – 9) + 16 28 + 16 Add. 44 Evaluate x ( y x – z y ) + x for x = 4, y = 2, and z = 3. y x ( y x – z y ) + x y
- 16. Check It Out! Example 3 60 – 7(7) Substitute 5 for x, 2 for y, and 60 for z. Simplify the powers. Subtract inside the parentheses. Multiply from left to right. 60 – 7(2 5 – 5 2 ) 60 – 7(32 – 25) 60 – 49 Subtract. 11 Evaluate z – 7(2 x – x y ) for x = 5, y = 2, and z = 60. z – 7(2 x – x y )
- 17. Additional Example 4: Geometry Application Simplify inside the parentheses. Multiply. Substitute the number of sides for n. Subtract inside the parentheses. 14 diagonals Use the expression ( n 2 – 3 n ) to find the number of diagonals in a 7-sided figure. (7 2 – 3 • 7) 1 2 (49 – 21) 1 2 ( n 2 – 3 n ) 1 2 (28) 1 2 1 2
- 18. A 7-sided figure has 14 diagonals. You can verify your answer by sketching the diagonals. Additional Example 4 Continued
- 19. Check It Out! Example 4 Simplify inside the parentheses. Multiply Substitute the number of sides for n. Subtract inside the parentheses. 2 diagonals Use the expression ( n 2 – 3 n ) to find the number of diagonals in a 4-sided figure. (4 2 – 3 • 4) 1 2 (16 – 12) 1 2 ( n 2 – 3 n ) 1 2 (4) 1 2 1 2
- 20. A 4-sided figure has 2 diagonals. You can verify your answer by sketching the diagonals. Check It Out! Example 4 Continued
- 21. Lesson Quiz Write in exponential form. 1. n • n • n • n 2. (–8) • (–8) • (–8) • ( h ) 256 – 213 (–8) 3 h 3. (–4) 4 5. Evaluate xz – y x for x = 5, y = 3, and z = 6. 6. A population of bacteria doubles in size every minute. The number of bacteria after 5 minutes is 15 2 5 . How many are there after 5 minutes? 480 Simplify. 4. n 4