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

Write

as

a

decimal.

3 4

means

3

?

4.

The

fraction

3 4

can

be

written

as

0.75,

since

3

?

4

=

0.75.

I can tell a fraction as a decimal will repeat by looking at it's _________________________.

Rule:

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)

Multiply each side by 10 because 1 digit repeats.

10N = 82.222...

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

?N = 8.222...

Subtract N = 8.222... to eliminate the repeating part.

9N = 74

10N ? 1N = 9N

9 74

9 =9

Divide each side by 9.

N

=

8

2 9

Simplify.

The decimal 8.2 can be written as 8 29.

A number is RATIONAL if...

Exercises Write each fraction or mixed number as a decimal.

1.

2 5

2.

-1

2 9

3.

7 8

4.

2

16 25

Determine if each number is rational or irrational.

5. 2.8

6. 2

7. 3300

8. 0.25256

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

9. 0.8

10. ?0.15

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, 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. 1

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

2. -16

Find the negative square root of 16; (?4)2 = 16, so - 16 = -4.

3. ?0.25 Find both square roots of 0.25; 0.52 = 0.25, so ?0.25 = ?0.5.

4. -49

There is no real square root because no number times itself is equal to ?49.

Example 5 Solve = . Check your solution(s).

2

=

4 9

Write the equation.

= ?4

9

Definition of square root

=

2 3

or

-

2 3

Check 2 ? 2 = 4 and (- 2) (- 2) = 4.

33 9

3

39

The

equation

has

two

solutions,

2 3

and

-

23.

Exercises Find each square root.

1. 4

2. 9

3. -49

4. -25

5. ?16

6. ? 0.64

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

9. 2 = 121

10. 2 = 3,600

11. 3

=

64 729

12. 2

=

121 196

13. 33 + 6 = 654

14. 43 - 5 = 11

15. 15 = 22 - 3

16. 162 = 0.52

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 to the nearest integer.

? The largest perfect square less than 204 is 196.

? The smallest perfect square greater than 204 is 225.

196 < 204 < 225 142 < 204 < 152

Write an inequality. 196 = 142 and 225 = 152.

142 < 204 < 152

Find the square root of each number.

14 < 204 < 15

Simplify.

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

Example 2 Estimate . 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 43 < 79.3 < 53

Write an inequality. 64 = 43 and 125 = 53.

364 < 379.3 < 3125

Find the cube root of each number.

4 < 379.3 < 5

Simplify.

So, 379.3 is between 4 and 5. Since 79.3 is closer to 64 than 125, the best whole number estimate for 379.3 is 4.

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

1. 8

2. 37

3. 14

4. 330

5. 3750

6. 3200

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 = 25

b. () 26(77) = (2 ? 7)(s6 ? s7) = 14(6+7) = 1413

The common base is 2. Add the exponents.

Commutative and Associative Properties The common base is s. 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

=

(((--22))160)

?

(5563)

?

(6632)

= (-2)4 ? 53 ? 61

= 16 ? 125 ? 6 or 12,000

Group by common base. Subtract the exponents. Simplify.

Exercises Simplify. Express using exponents.

1. 52 ? 56

2. 2 ? 7

3. 25 ? 6

4. 42 ? (-56)

5.

79 73

9.

25 ? 37 ? 43 21 ? 35 ? 4

6.

14 6

10.

415 412

? ?

(-5)6 (-5)4

7.

157 52

11.

67 65

? ?

76 75

? ?

85 84

8. 108

2

12.

(-3)6 (-3)4

? ?

105 103

Unit 8.1 REVIEW

Name ___________________________

Lesson 5 Reteach 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 (-).

(-324)3 = (-3)3 ? 2 ? 3 ? 4 ? 3

= -27612

Power of a product Simplify.

Exercises Simplify.

1. (43)5

2. (42)7

3. (92)4

4. (542)5

5. (322)6

6. (7437)2

7. (-435)2

8. (-549)7

9. (0.28)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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