1. Uses of Powers
Lesson 2.5.3

2. Uses of Powers 2.5.3 California Standards: What it means for you:
Lesson
2.5.3
Uses 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 1.2
Add, 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

3. Lesson
2.5.3
Uses 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.

4. Uses of Powers 2.5.3 Exponents are Used in Some Formulas
Lesson
2.5.3
Uses of Powers
Exponents are Used in Some Formulas
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.

5. Lesson
2.5.3
Uses of Powers
The formula for the area of a square is Area = s • s = s2, where s represents the side length of the square.
42 = 16 4 3
32 = 9 2
22 = 4 1
12 = 1
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.

6. Uses of Powers 2.5.3 Find the area of the square shown. Solution
Lesson
2.5.3
Uses of Powers
Example 1
Find the area of the square shown.
1 cm
Solution
Each small square is 1 cm wide. So the side length of the whole square is 2 cm.
The area of the whole square is 2 cm • 2 cm = 4 cm2.
You can see that this is true, because it is made up of four smaller 1 cm2 squares.
Solution follows…

7. Lesson
2.5.3
Uses of Powers
Example 2
A square has a side length of 11 inches. Find its area.
Solution
Area = (side length)2
Area = 112 = 11 • 11 = 121
Units: inches • inches = in2
Area = 121 in2
Solution follows…

8. Uses of Powers 2.5.3 Guided Practice
Lesson
2.5.3
Uses of Powers
Guided Practice
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 = m2
(5 • 2)2 = 102 = 10 • 10 = 100 mm2
Solution follows…

9. Lesson
2.5.3
Uses of Powers
Exponents are Used to Find the Volume of Some Solids
Exponents are also used in formulas to work out volumes of solids, like cubes, spheres, and prisms. s
The formula for the volume of a cube is:
Volume = s • s • s = s3,
where s represents the side length of the cube.

10. Lesson
2.5.3
Uses of Powers
23 = 8
13 = 1 1 2
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.

11. Lesson
2.5.3
Uses of Powers
Example 3
A cube has a side length of 5 cm. Find its volume.
Solution
Volume = (side length)3
Volume = 5 • 5 • 5 = 53 = 125
Units: cm • cm • cm = cm3
Volume = 125 cm3
Solution follows…

12. Uses of Powers 2.5.3 Guided Practice
Lesson
2.5.3
Uses of Powers
Guided Practice
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.
4 m
s3 = 4 • 4 • 4 = 64 m3
s3 = 5 • 5 • 5 = 125 feet3
s3 = 7.5 • 7.5 • 7.5 = mm3
1 in
s3 = 2 • 2 • 2 = 8 in3
Solution follows…

13. Uses of Powers 2.5.3 Use Scientific Notation to Write Big Numbers
Lesson
2.5.3
Uses of Powers
Use Scientific Notation to Write Big Numbers
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.

14. Uses of Powers 2.5.3 For instance: 570,000,000 = 5.7 × 100,000,000
Lesson
2.5.3
Uses of Powers
For instance:
570,000,000 = 5.7 × 100,000,000
5.7 × 100,000,000 = 5.7 × 108.
The second factor is a power of ten.
You can write it in base and exponent form.
So 5.7 × 108 is 570,000,000 written in scientific notation.

15. Uses of Powers 2.5.3 Scientific Notation
Lesson
2.5.3
Uses of Powers
Scientific Notation
To write a number in scientific notation turn it into two factors:
— the first factor must be a number that’s at least one but less than ten.
— the second factor must be a power of 10 written in exponent form.
For example: × 103, × × 10–5

16. Lesson
2.5.3
Uses of Powers
Example 4
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
Write the number as a product of the two factors.
= 1.28 × 1011
Solution follows…

17. Lesson
2.5.3
Uses of Powers
Example 5
The number 5.1× 107 is written in scientific notation.
Write it out in full.
Solution
Write out the power of ten as a factor in full.
5.1 × 107 = 5.1 × 10,000,000
Multiply the two together: move the decimal point as many places to the right as there are zeros in the power of ten.
= 51,000,000
Solution follows…

18. Uses of Powers 2.5.3 Guided Practice
Lesson
2.5.3
Uses of Powers
Guided Practice
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
Solution follows…

19. Uses of Powers 2.5.3 Guided Practice
Lesson
2.5.3
Uses of Powers
Guided Practice
Write out the numbers in Exercises 13–16 in full.
× 103
× 106
15. –5.05 × 107
× 109
3100
8,140,000
–50,500,000
9,091,000,000
Solution follows…

20. Uses of Powers 2.5.3 Independent Practice
Lesson
2.5.3
Uses of Powers
Independent Practice
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
Solution follows…

21. Uses of Powers 2.5.3 Independent Practice
Lesson
2.5.3
Uses of Powers
Independent Practice
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
Solution follows…

22. Uses of Powers 2.5.3 Independent Practice
Lesson
2.5.3
Uses of Powers
Independent Practice
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
Solution follows…

23. Uses of Powers 2.5.3 Independent Practice
Lesson
2.5.3
Uses of Powers
Independent Practice
Write out the numbers in Exercises 15–18 in full.
× 105
× 108
17. –9.1 × 104
× 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
Solution follows…

24. Lesson
2.5.3
Uses of Powers
Round Up
When you’re finding the area of a square or volume of a cube, your calculation will always involve powers.
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.
Powers also come in useful for writing very large numbers in a shorter form — that’s what scientific notation is for.