Lesson 1 Reteach Rational Numbers - Mrs. Pierce

Unit 8.1 REVIEW

Name ___________________________

Lesson 1 Reteach Rational Numbers

To express a fraction as a decimal, divide the numerator by the denominator.

Example 1

I can tell a fraction as a decimal will repeat by

looking at it¡¯s _________________________.

?

Write ? as a decimal.

3

4

means 3 ¡Â 4.

Rule:

3

The fraction 4 can be written as 0.75, since 3 ¡Â 4 = 0.75.

Example 2

Write 8.? as a mixed number in simplest form.

Assign a variable to the value 8.2 Let N = 8.222¡­ . Then perform the operations on N to determine its value.

N = 8.2? or 8.222¡­.

10(N) = 10(8.222)

10N = 82.222¡­

Multiply each side by 10 because 1 digit repeats.

Multiplying by 10 moves the decimal point 1 place to the right.

¨CN = 8.222¡­

Subtract N = 8.222¡­ to eliminate the repeating part.

9N = 74

10N ¨C 1N = 9N

9?

9

=

74

9

2

N = 89

Divide each side by 9.

A number is RATIONAL if¡­

Simplify.

2

The decimal 8.2 can be written as 8 9.

Exercises

Write each fraction or mixed number as a decimal.

2

1. 5

2

7

2. ?1 9

16

4. 2 25

3. 8

Determine if each number is rational or irrational.

5. 2.8?

6. 2?

3

7. ¡Ì300

8. 0.25256

Write each decimal as a fraction or mixed number in simplest form.

9. 0.8

????

10. ¨C0.15

11. 0.1

12. 1.7

Unit 8.1 REVIEW

Name ___________________________

Lesson 2 Reteach

Roots

A square root of a number is one of its two equal factors. A radical sign, ¡Ì is used to indicate a positive square root.

Every positive number has both a negative and positive square root.

Examples

Find each square root.

1. ¡Ì1

Find the positive square root of 1; 12 = 1, so ¡Ì1 = 1.

2. ?¡Ì16

Find the negative square root of 16; (¨C4)2 = 16, so ? ¡Ì16 = ?4.

3. ¡À¡Ì0.25

Find both square roots of 0.25; 0.52 = 0.25, so ¡À¡Ì0.25 = ¡À0.5.

4. ¡Ì?49

There is no real square root because no number times itself is equal to ¨C49.

Example 5

?

?

Solve ?? = . Check your solution(s).

4

?2 = 9

Write the equation.

4

? = ¡À¡Ì9

2

Definition of square root

2

? = 3 or ? 3

2

2

3

3

Check ?

2

3

4

2

2

4

9

3

3

9

= and (? ) (? ) = .

The equation has two solutions, and

2

? 3.

Exercises

Find each square root.

1. ¡Ì4

2. ¡Ì9

3. ?¡Ì49

4. ?¡Ì25

5. ¡À¡Ì16

6. ¨C ¡Ì0.64

ALGEBRA Solve each equation. Show each step used to do so. Use ¡À in your answer, when applicable.

9. ? 2 = 121

12. ? 2 =

121

196

15. 15 = 2?2 ? 3

64

729

10. ?2 = 3,600

11. ?3 =

13. 3? 3 + 6 = 654

14. 4?3 ? 5 = 11

16. 162 = 0.5? 2

Unit 8.1 REVIEW

Name ___________________________

Lesson 3 Reteach

Approximating Roots

Most numbers are not perfect squares or cubes. You can estimate roots for these numbers.

Example 1

Estimate ¡Ì??? to the nearest integer.

? The largest perfect square less than 204 is 196.

? The smallest perfect square greater than 204 is 225.

196 < 204 < 225

Write an inequality.

142 < 204 < 152

196 = 142 and 225 = 152.

¡Ì142 < ¡Ì204 < ¡Ì152

14 < ¡Ì204 < 15

Find the square root of each number.

Simplify.

So, ¡Ì204 is between 14 and 15. Since 204 is closer to 196 than 225, the best whole number estimate for ¡Ì204 is 14.

Example 2

?

Estimate ¡Ì??. ? to the nearest integer.

? The largest perfect cube less than 79.3 is 64.

? The smallest perfect cube greater than 79.3 is 125.

64 < 79.3 < 125

Write an inequality.

43 < 79.3 < 53

64 = 43 and 125 = 53.

3

3

3

¡Ì64 < ¡Ì79.3 < ¡Ì125

3

4 < ¡Ì79.3 < 5

Find the cube root of each number.

Simplify.

3

3

So, ¡Ì79.3 is between 4 and 5. Since 79.3 is closer to 64 than 125, the best whole number estimate for ¡Ì79.3 is 4.

Exercises

Estimate square roots to the nearest hundredth and cube roots to the nearest tenth (¡À ?. ?). Use graph paper if

necessary.

1. ¡Ì8

3

4. ¡Ì30

2. ¡Ì37

3

5. ¡Ì750

3. ¡Ì14

3

6. ¡Ì200

Unit 8.1 REVIEW

Name ___________________________

Lesson 4 Reteach

Multiply and Divide Powers

The Product of Powers rule states that to multiply powers with the same base, add their exponents.

Example 1

Simplify. Express using exponents.

a. ?? ? ??

23 ? 22 = 23+2

The common base is 2.

= 25

?

Add the exponents.

?

b. ?? (?? )

2? 6 (7? 7 ) = (2 ? 7)(s 6 ? s 7 )

Commutative and Associative Properties

= 14(? 6+7 )

The common base is s.

= 14?13

Add the exponents.

The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.

Example 2

Simplify

??

.

?

Express using exponents.

?8

?1

= ? 8?1 The common base is k.

= ?7

Subtract the exponents.

Example 3

Simplify

(??)?? ? ?? ? ??

(??)? ? ?? ? ??

(?2)10 ? 56 ? 68

(?2)6 ? 58 ? 62

.

(?2)10

56

63

= ( (?2)6 ) ? (53 ) ? (62 )

Group by common base.

= (?2)4 ? 53 ? 61

Subtract the exponents.

= 16 ? 125 ? 6 or 12,000

Simplify.

Exercises

Simplify. Express using exponents.

1. 52 ? 56

5.

9.

79

73

25 ? 37 ? 43

21 ? 35 ? 4

2. ? 2 ? ? 7

6.

? 14

?6

415 ? (?5)6

10. 412 ? (?5)4

3. 2?5 ? 6?

7.

15? 7

5? 2

67 ? 76 ? 85

11. 65 ? 75 ? 84

4. 4? 2 ? (?5? 6 )

8.

10?8

2?

(?3)6 ? 105

12. (?3)4 ? 103

Unit 8.1 REVIEW

Lesson 5 Reteach

Name ___________________________

Powers of Powers

Power of a Power: To find the power of a power, multiply the exponents.

Power of a Product: To find the power of a product, find the power of each factor and multiply.

Example 1

Simplify (?? )? .

(53 )6 = 53 ? 6

= 518

Power of a power

Simplify.

Example 2

Simplify (???? ?? )? .

(?3?2 ?4 )3 = (?3)3 ? ?2 ? 3 ? ?4 ? 3

= ?27?6 ?12

Power of a product

Simplify.

Exercises

Simplify.

1. (43 )5

2. (42 )7

3. (92 )4

4. (5? 4 ? 2 )5

5. (3? 2 ? 2 )6

6. (7?4 ?3 ? 7 )2

7. (?4?3 ? 5 )2

8. (?5?4 ?9 )7

9. (0.2? 8 )2

Apply the Converse to Power of a Power Property. Write an equivalent expression with only 1 exponent (after

parenthesis).

10. -36h2g4

11. 125c6k9

12. 10,000p4a8

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