8-7 Powers and Roots

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpSimplify.

1. 62

2. 72

3. 112

4. 152

36

49

121

Course 2

8-7 Powers and Roots

225

Problem of the Day

The square of two whole numbers are 16 units apart on a number line. What are the two numbers?3 and 5

Course 2

8-7 Powers and Roots

Learn to express and evaluate numbers using powers and roots.

Course 2

8-7 Powers and Roots

Vocabulary

perfect squaresquare rootradical sign

Insert Lesson Title Here

Course 2

Powers and Roots8-7

Course 2

Powers and Roots8-7

2Exponent

Base

Recall that a power is a number represented by a base and an exponent. The exponent tells you how many times to use the base as a repeated factor.

A square with sides that measure 3 units each has an area of 3 · 3, or 32. Notice that the area of the square is represented by a power in which the base is the side length and the exponent is 2. A power in which the exponent is 2 is called a square.

Model each power using a square. Then evaluate the power.

Additional Example 1A & 1B: Finding Squares

Course 2

8-7 Powers and Roots

A. 122

A = lwA = 12 · 12A = 144The square of 12 is 144.

Substitute.Multiply.

B. (3.6)2

A = lwA = 3.6 · 3.6A = 12.96The square of 3.6 is 12.96.

Substitute.Multiply.

12

12

3.6

3.6

Try This: Example 1A &1B

Insert Lesson Title Here

Course 2

8-7 Powers and Roots

A. 102

A = lwA = 10 · 10A = 100The square of 10 is 100.

Substitute.Multiply.

B. (5.2)2

A = lwA = 5.2 · 5.2A = 27.04The square of 5.2 is 27.04.

Substitute.Multiply.

Model each power using a square. Then evaluate the power.10

10

5.2

5.2

Course 2

8-7 Powers and Roots

A perfect square is the square of a whole number. The number 49 is a perfect square because 49 =72 and 7 is a whole number. The number 6.25 is not a perfect square.

The square root of a number is one of the two equal factors of the number. Four is a square root of 16 because 4 · 4 = 16. The symbol for a square root is √ , which is called a radical sign.

Course 2

8-7 Powers and Roots

√16 = 4 is read as “The square root of 16 is 4.”

Reading Math

Most calculators have square-root keys that you can use to quickly find approximate square roots of nonperfect squares. You can also use perfect squares to estimate the square roots of nonperfect squares.

Estimate each square root to the nearest whole number. Use a calculator to check your answer.

Additional Example 2A: Estimating Square Roots

Course 2

8-7 Powers and Roots

36 < 40 < 49

Check

Find the perfect squares nearest 40.

Find the square roots of 36 and 49.

40 is nearer in value to 36 than to 49.

6 is a reasonable estimate.

√36 < √40 < 49√

6 < 40 < 7√

40 6√

40 6.32455532033√ Use a calculator to approximate √40.

A. √40

Additional Example 2B: Estimating Square Roots

Course 2

8-7 Powers and Roots

Estimate each square root to the nearest whole number. Use a calculator to check your answer.

64 < 79 < 81

Check

Find the perfect squares nearest 79.

Find the square roots of 64 and 81.

79 is nearer in value to 81 than to 64.

B. √79

79 9√

79 8.8881944√

8 < 79 < 9√

√64 < √79 < 81√

Use a calculator to approximate √79.

9 is a reasonable estimate.

Try This: Example 2A

Insert Lesson Title Here

Course 2

8-7 Powers and Roots

Estimate each square root to the nearest whole number. Use a calculator to check your answer.

A. √22

16 < 22 < 25

Check

Find the perfect squares nearest 22.

Find the square roots of 16 and 25.

22 is nearer in value to 25 than to 16.

4 < 22 < 5√

22 5√

22 4.690415759√

√16 < √22 < 25√

Use a calculator to approximate√22.

5 is a reasonable estimate.

Try This: Example 2B

Insert Lesson Title Here

Course 2

8-7 Powers and Roots

B. √53

49 < 53 < 64

Check

Find the perfect squares nearest 53.

Find the square roots of 49 and 64.

53 is nearer in value to 49 than to 64.

7 < 53 < 8√

53 7√

53 7.2801098828√

√49 < √53 < 64√

Use a calculator to approximate√53.

Estimate each square root to the nearest whole number. Use a calculator to check your answer.

7 is a reasonable estimate.

A Coast Guard boat searching for a lost sailboat covers a square area of 125 mi2. What is the approximate length of each side of the square area? Round your answer to the nearest mile.

Additional Example 3: Recreation Application

Course 2

8-7 Powers and Roots

121 < 125 < 144 Find the perfect squares nearest 125.

Find the square roots of 121 and 144.

Each side of the search area is about 11 miles long.

125 is nearer in value to 121 than to 144.

The length of each side of the square is √125 .

< < √125√121 √144

11 < < 12√125

√125 11

Try This: Example 3

A tent was advertised in the newspaper as having an enclosed square area of 168 ft2. What is the approximate length of the sides of the square area? Round your answer to the nearest foot.

Insert Lesson Title Here

Course 2

8-7 Powers and Roots

The length of each side of the square is √168 .144 < 168 < 169 Find the perfect squares nearest 168.

Find the square roots of 144 and 169.

< < √168√144 √169

12 < < 13√168

√168 13

Each side of the tent is about 13 feet long.

168 is nearer in value to 169 than to 144.

Lesson Quiz

Evaluate each power.

1. 162

2. (3.5)2

Estimate each square root to the nearest whole number. Use a calculator to check the reasonableness of your answers.

3.

5. A square dining room table has an area of 20 ft2.

What is the length of each side of the table, to the nearest tenth?

12.25

256

Insert Lesson Title Here

4 7

Course 2

8-7 Powers and Roots

√15 4. √52

4.5 ft