Powers of Rational Numbers Lesson 2.5.2

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

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

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

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.

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…

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…

Powers of Rational Numbers Lesson 2.5.2 Powers of Rational Numbers Guided Practice Evaluate the exponential expressions in Exercises 1–6. 1. 2. 3. 4. 5. 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…

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.

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.

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…

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 = 0.0529 529 10,000 Divide to give a decimal Solution follows…

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)2 8. (0.2)3 9. (0.78)1 10. (0.12)2 11. (0.15)3 12. (0.08)2 0.5 × 0.5 = 0.25 0.2 × 0.2 × 0.2 = 0.008 0.78 0.12 × 0.12 = 0.0144 0.15 × 0.15 × 0.15 = 0.003375 0.08 × 0.08 = 0.0064 Solution follows…

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. 1. • • 2. 0.25 • 0.25 3. • • • 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…

Powers of Rational Numbers Lesson 2.5.2 Powers of Rational Numbers Independent Practice Evaluate the exponential expressions in Exercises 5–10. 5. 6. 7. 8. 9. 10. 1 9 2 4 3 8 5 – 81 1 64 1 27 8 8 9 3125 1 125 8 – Solution follows…

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…

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)2 13. (0.1)4 14. (0.21)2 15. (0.97)1 16. (0.02)2 17. (0.25)3 0.16 0.0001 0.0441 0.97 0.0004 0.015625 Solution follows…

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.