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Lesson 2.5.3Lesson 2.5.3

Uses of PowersUses of Powers

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Lesson

2.5.3Uses of PowersUses of Powers

California Standards:Number Sense 1.1

Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general.

Number Sense 1.2Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.

What it means for you:You’ll see how you can use exponents to work out areas of squares and volumes of cubes, and learn about a shorter way to write very large numbers.

Key words:• squared• cubed• scientific notation

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2.5.3Uses of PowersUses of Powers

You’ll come across powers a lot both in math and real-life situations. That’s because you use them to work out areas and volumes.

They’re also handy when you need to write out a very big number — you can use powers to write these numbers in a shorter form.

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Exponents are Used in Some Formulas

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2.5.3Uses of PowersUses of Powers

Exponents are used in the formulas for the areas of squares and circles.

In this Lesson you’ll see how exponents are used in finding the area of a square.

In the next Chapter you’ll use a formula to find the area of a circle.

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2.5.3Uses of PowersUses of Powers

The formula for the area of a square is Area = s • s = s2, where s represents the side length of the square.

When you find the area of a square, the side length is used as a factor twice in the multiplication. So raising a number to the second power is called squaring it.

42 = 16

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4

3

3

32 = 9

2

2

22 = 4

1

1

12 = 1

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Example 1

Solution follows…

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2.5.3Uses of PowersUses of Powers

Find the area of the square shown.

1 cmSolution

The area of the whole square is 2 cm • 2 cm = 4 cm2.

Each small square is 1 cm wide. So the side length of the whole square is 2 cm.

You can see that this is true, because it is made up of four smaller 1 cm2 squares.

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Example 2

Solution follows…

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2.5.3Uses of PowersUses of Powers

A square has a side length of 11 inches. Find its area.

Solution

Units: inches • inches = in2

Area = (side length)2

Area = 112 = 11 • 11 = 121

Area = 121 in2

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Guided Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Find the areas of the squares in Exercises 1–4.1.

2. Square of side length 6 feet.

3. Square of side length 3.5 m.

4.

3 miles

5 mm

s2 = 3 • 3 = 9 miles2

s2 = 6 • 6 = 36 feet2

s2 = 3.5 • 3.5 = 12.25 m2

(5 • 2)2 = 102 = 10 • 10 = 100 mm2

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Exponents are Used to Find the Volume of Some Solids

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2.5.3Uses of PowersUses of Powers

Exponents are also used in formulas to work out volumes of solids, like cubes, spheres, and prisms.

The formula for the volume of a cube is:

Volume = s • s • s = s3,

where s represents the side length of the cube.

s

s

s

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When you find the volume of a cube, the side length is used as a factor three times in the multiplication. So raising a number to the third power is called cubing it.

23 = 813 = 1

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1

1

2

2

2

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Example 3

Solution follows…

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2.5.3Uses of PowersUses of Powers

A cube has a side length of 5 cm. Find its volume.

Solution

Units: cm • cm • cm = cm3

Volume = (side length)3

Volume = 5 • 5 • 5 = 53 = 125

Volume = 125 cm3

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Find the volumes of the cubes in Exercises 5–8. 5.

6. Cube of side length 5 feet.

7. Cube of side length 7.5 mm.

8.

Guided Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

4 m

4 m

4 m

1 in

s3 = 4 • 4 • 4 = 64 m3

s3 = 5 • 5 • 5 = 125 feet3

s3 = 7.5 • 7.5 • 7.5 = 421.875 mm3

s3 = 2 • 2 • 2 = 8 in3

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Use Scientific Notation to Write Big Numbers

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2.5.3Uses of PowersUses of Powers

Sometimes in math and science you’ll need to deal with numbers that are very big, like 570,000,000.

To avoid having to write numbers like this out in full every time, you can rewrite them as a product of two factors.

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For instance:

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2.5.3Uses of PowersUses of Powers

The second factor is a power of ten.

So 5.7 × 108 is 570,000,000 written in scientific notation.

5.7 × 100,000,000 = 5.7 × 108.

570,000,000 = 5.7 × 100,000,000

You can write it in base and exponent form.

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Scientific Notation

To write a number in scientific notation turn it into two factors:

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2.5.3Uses of PowersUses of Powers

— the second factor must be a power of 10 written in exponent form.

— the first factor must be a number that’s at least one but less than ten.

For example: 3.1 × 103, 0.4 × 107 9.9 × 10–5

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Example 4

Solution follows…

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2.5.3Uses of PowersUses of Powers

Write the number 128,000,000,000 in scientific notation.

Solution

Split the number into a decimal between 1 and 10 and a power of ten.

128,000,000,000 = 1.28 × 100,000,000,000

= 1.28 × 1011

Write the number as a product of the two factors.

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Example 5

Solution follows…

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2.5.3Uses of PowersUses of Powers

The number 5.1× 107 is written in scientific notation. Write it out in full.

Solution

5.1 × 107 = 5.1 × 10,000,000

= 51,000,000

Write out the power of ten as a factor in full.

Multiply the two together: move the decimal point as many places to the right as there are zeros in the power of ten.

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Guided Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Write the numbers in Exercises 9–12 in scientific notation.

9. 6,700,000

10. 32,800

11. –270,000

12. 1,040,000,000

6.7 × 106

3.28 × 104

–2.7 × 105

1.04 × 109

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Guided Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Write out the numbers in Exercises 13–16 in full.

13. 3.1 × 103

14. 8.14 × 106

15. –5.05 × 107

16. 9.091 × 109

3100

8,140,000

–50,500,000

9,091,000,000

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Independent Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Find the areas of the squares in Exercises 1–4.

1. Square of side length 2 cm.

2. Square of side length 8 yd.

3. Square of side length 13 m.

4. Square of side length 5.5 ft.

5. Maria is painting a wall that is 8 feet high and 8 feet wide.

She has to apply two coats of paint. Each paint can will cover 32 feet2. How many cans of paint does she need?

4 cm2

64 yd2

169 m2

30.25 ft2

4 cans

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Independent Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Find the volumes of the cubes in Exercises 6–9.

6. Cube of side length 3 ft.

7. Cube of side length 6 yd.

8. Cube of side length 9 cm.

9. Cube of side length 1.5 in.

10. Tyreese is tidying up his little sister’s toys. Her building blocks are small cubes, each with a side length of 3 cm. They completely fill a storage box that is a cube with a side length of 15 cm. How many blocks does Tyreese’s sister have?

27 ft3

216 yd3

729 cm3

3.375 in3

125 blocks

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Independent Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Write the numbers in Exercises 11–14 in scientific notation.

11. 21,000

12. –51,900,000

13. 42,820,000

14. 31,420,000,000,000

2.1 × 104

–5.19 × 107

4.282 × 107

3.142 × 1013

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Independent Practice

Solution follows…

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2.5.3Uses of PowersUses of Powers

Write out the numbers in Exercises 15–18 in full.

15. 8.4 × 105

16. 2.05 × 108

17. –9.1 × 104

18. 3.0146 × 1010

19. In 2006 the population of the USA was approximately 299,000,000. Of those 152,000,000 were female. How many were male? Give your answer in scientific notation.

840,000

205,000,000

–91,000

30,146,000,000

1.47 × 108

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Round UpRound Up

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2.5.3Uses of PowersUses of Powers

When you’re finding the area of a square or volume of a cube, your calculation will always involve powers.

Powers also come in useful for writing very large numbers in a shorter form — that’s what scientific notation is for.

That’s because the formulas for both the area of a square and the volume of a cube involve repeated multiplication of the side length.