1. Pre-Algebra 2-7 Properties of Exponents

2. Evaluate. 27 1. 3 3 2. 4 4 4 4 3. b 2 for b = 4 4. n 2 r for n = 3 and r = 2 256 16 18 Warm Up

3. Learn to apply the properties of exponents and to evaluate the zero exponent.

4. The factors of a power, such as 7 4, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 7 7 7 = 7 4 (7 7 7) 7 = 7 3 7 1 = 7 4 (7 7) (7 7) = 7 2 7 2 = 7 4

5. Words NumbersAlgebra To multiply powers with the same base, keep the base and add the exponents. b m b n = b m + n 3 5 3 8 = 3 5 + 8 = 3 13 MULTIPLYING POWERS WITH THE SAME BASE

6. A. 6 6 6 3 6 9 6 6 + 3 B. n 5 n 7 n 12 n 5 + 7 Add exponents. Multiply. Write the product as one power. Example 1A & 1B: Multiplying Powers with the Same Base

7. D. 24 4 24 4 C. 2 5 2 2 6 2 5 + 1 24 8 4 + 4 Think: 2 = 2 1 Multiply. Write the product as one power. Add exponents. Example 1: Multiplying Powers with the Same Base Continued

8. A. 4 2 4 4 4 6 4 2 + 4 B. x 2 x 3 x 5 x 2 + 3 Add exponents. Multiply. Write the product as one power. Try This

9. D. 41 2 41 7 C. x 5 y 2 41 9 2 + 7 Multiply. Write the product as one power. Cannot combine; the bases are not the same. Add exponents. x 5 y 2 Try This

10. Notice what occurs when you divide powers with the same base. DIVIDING POWERS WITH THE SAME BASE WordsNumbersAlgebra To divide powers with the same base, keep the base and subtract the exponents. 6 5 6 9 – 4 6 9 6 4 = = b m – n b m b n = 5 5 5353 = 5  5  55  5  5 5  5  5  5  5 = 5 5 = 5 2 = 5  5  55  5  5 5  5  5  5  5

11. Subtract exponents. 7 2 7 5 – 3 7 5 7 3 Divide. Write the product as one power. A. x 10 x 9 B. Subtract exponents. x 10 – 9 x Think: x = x 1 Example 2: Dividing Powers with the Same Base

12. Subtract exponents. 9797 9 9 – 2 9 9 9 2 Divide. Write the product as one power. A. B. e 10 e 5 Subtract exponents. e 10 – 5 e 5 Try This

13. When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent. This result can be confirmed by writing out the factors. 1 = 4 2 4 2 4 2 – 2 = 4 0 = 1 = = (4 4) = 1 1 1 = 4 2 2 = (4 4) 4

14. 0 0 does not exist because 0 0 represents a quotient of the form But the denominator of this quotient is 0, which is impossible, since you cannot divide by 0. Helpful Hint 0n0n0n0n.

15. THE ZERO POWER WordsNumbers Algebra The zero power of any number except 0 equals 1. 100 0 = 1 (–7) 0 = 1 a = 1, if a  0

16. A light-year, or the distance light travels in one year, is almost 10 18 centimeters. To convert this number to kilometers, you must divide by 10 5. How many kilometers is a light-year? 10 18 - 5 A light-year is almost 10 13 km. 10 13 10 18 10 5 Subtract exponents. Example 3: Astronomy Application

17. A ship has 10 7 kilograms of grain loaded into its cargo hold. A metric ton is 10 3 kilograms. How many metric tons of grain were loaded? 10 7 - 3 The ship had 10 4 metric tons of grain loaded. 10 4 7 3 Subtract exponents. The weight in metric tons is equal to the weight in kilograms divided by 10 kilograms per metric ton. 3 Try This

18. Write the product or quotient as one power 3. 8 9 n 7 1. n 3  n 4 10 9 10 5 10 4 4. t 2 5. 3 3 3 2 3 5 3 10 2. 8 8 8 t 9 t 7 Lesson Quiz: Part 1

19. A school would like to purchase new globes. They can get six dozen for $705.80 from Company A. From Company B, they can buy a half gross for $725.10. Which company should they buy from? (1 gross = 12 2 items) Company A 6. Lesson Quiz: Part 2