1. Powers of Rational Numbers
Lesson 2.5.2

2. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
California Standards:
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.
Algebra and Functions 2.1
Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
What it means for you:
You’ll learn how to take fractions and decimals to powers.
Key words:
power
exponent
base
decimal
fraction

3. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
In the same way as you can raise whole numbers to powers, you can also raise fractions and decimals to powers. 1 4 5 2 3
0.52
0.43
0.255

4. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
You Can Raise a Fraction to a Power
A fraction raised to a power means exactly the same as a whole number raised to a power — repeated multiplication. But now the complete fraction is the base. 2 3
Base
Exponent
This expression means • . 2 3

5. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
When you raise a fraction to a power, you are raising the numerator and the denominator separately to the same power.
For example: 2 3 = 22 32
This makes evaluating the fraction easier. You can evaluate the numerator and the denominator separately.

6. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Example 1
Evaluate 1 4 3
Solution 1 4 3 = 13 43
Raise both the numerator and the denominator to the third power.
13 = 1 • 1 • 1 = 1
Evaluate each power separately
43 = 4 • 4 • 4 = 64 = 13 43 1 64
Write the powers as a fraction
Solution follows…

7. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Example 2
Evaluate – 2 5
Solution
– =
(–2)2 52 2 5
Raise both the numerator and the denominator to the second power.
(–2)2 = –2 • –2 = 4
Evaluate each power separately
52 = 5 • 5 = 25 = 4 25
(–2)2 52
Write the powers as a fraction
Solution follows…

8. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Guided Practice
Evaluate the exponential expressions in Exercises 1–6. 1 2 4 5 3 9 7 10 – 4 1 22 12 = 16 1 24 14 = 3 5 31 51 =
343
729 73 93 =
100 9
102
–32 =
1000 27 –
103
–33 =
Solution follows…

9. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
You Can Raise a Decimal to a Power
A decimal raised to a power means exactly the same as a whole number raised to a power — it’s a repeated multiplication. The decimal is the base.
0.242
Base
Exponent
This expression is the same as saying 0.24 • 0.24.

10. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
When you evaluate a decimal raised to a power, you multiply the decimal by itself the specified number of times.
The tricky thing when you’re multiplying decimals is to get the decimal point in the right place — you saw how to do this in Section 2.4.

11. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Example 3
Evaluate (0.3)3.
Solution
The multiplication you are doing here is 0.3 • 0.3 • 0.3.
(0.3)3 = × × 3 10
Write the decimals as fractions =
3 × 3 × 3
10 × 10 × 10
Multiply the fractions
= = 27 ÷ 1000 = 0.027 27
1000
Divide to give a decimal
Solution follows…

12. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Example 4
Evaluate (0.23)2.
Solution
The multiplication you are doing here is 0.23 • 0.23.
(0.23)2 = × 23
100
Write the decimals as fractions =
23 × 23
100 × 100
Multiply the fractions
= = 529 ÷ 10,000 =
529
10,000
Divide to give a decimal
Solution follows…

13. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Guided Practice
Evaluate the exponential expressions in Exercises 7–12.
7. (0.5) (0.2)3
9. (0.78) (0.12)2
11. (0.15) (0.08)2
0.5 × 0.5 = 0.25
0.2 × 0.2 × 0.2 = 0.008
0.78
0.12 × 0.12 =
0.15 × 0.15 × 0.15 =
0.08 × 0.08 =
Solution follows…

14. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Independent Practice
Write each of the expressions in Exercises 1–4 in base and exponent form.
• • • 0.25
• • • 4. – • – • – 1 2 1 2 1 2 2 1 3
0.252 1 7 1 7 1 7 1 7 1 4 1 4 1 4 7 1 4 4 1 3 –
Solution follows…

15. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Independent Practice
Evaluate the exponential expressions in Exercises 5–10. 1 9 2 4 3 8 5 – 81 1 64 1 27 8 8 9
3125 1
125 8 –
Solution follows…

16. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Independent Practice
11. Mark is feeding chickens. He divides 135 g of corn into thirds. Each portion is then divided into thirds again to give small portions.
What fraction of the original amount is in each small portion?
How much does each small portion weigh? 9 1
15 g
Solution follows…

17. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Independent Practice
Evaluate the exponential expressions in Exercises 12–17.
12. (0.4) (0.1)4
14. (0.21) (0.97)1
16. (0.02) (0.25)3
0.16
0.0001
0.0441
0.97
0.0004
Solution follows…

18. Powers of Rational Numbers
Lesson
2.5.2
Powers of Rational Numbers
Round Up
To raise a fraction to a power, you raise the numerator and the denominator separately to the same power.
To raise a decimal to a power, you use the decimal as the base and raise it to a power as you would a whole number — just by multiplying it by itself the correct number of times.