1. 2-7 Properties of Exponents Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Warm Up Evaluate. Course 3 2-7 Properties of Exponents 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

3. Problem of the Day Calculate 6 to the fourth power minus 56. Course 3 2-7 Properties of Exponents 1240

4. Learn to apply the properties of exponents and to evaluate the zero exponent. Course 3 2-7 Properties of Exponents

5. The factors of a power, such as 7 4, can be grouped in different ways. Notice the relationship of the exponents in each product. Course 3 2-7 Properties of Exponents 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

6. Course 3 2-7 Properties of Exponents 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

7. Additional Example 1A & 1B: Multiplying Powers with the Same Base Course 3 2-7 Properties of Exponents 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.

8. D. 24 4 24 4 C. 2 5 2 Course 3 2-7 Properties of Exponents 2 6 2 5 + 1 24 8 4 + 4 Think: 2 = 2 1 Additional Example 1: Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. Add exponents.

9. Try This: Example 1A & 1B Course 3 2-7 Properties of Exponents 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.

10. D. 41 2 41 7 C. x 5 y 2 Course 3 2-7 Properties of Exponents 41 9 2 + 7 Try This: Example 1C & 1D Multiply. Write the product as one power. Cannot combine; the bases are not the same. Add exponents. x 5 y 2

11. Course 3 2-7 Properties of Exponents 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

12. Course 3 2-7 Properties of Exponents Subtract exponents. 7 2 7 5 – 3 7 5 7 3 Additional Example 2: Dividing Powers with the Same Base Divide. Write the product as one power. A. x 10 x 9 B. Subtract exponents. x 10 – 9 x Think: x = x 1

13. Course 3 2-7 Properties of Exponents Subtract exponents. 9797 9 9 – 2 9 9 9 2 Try This: Example 2 Divide. Write the product as one power. A. B. e 10 e 5 Subtract exponents. e 10 – 5 e 5

14. When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent. Course 3 2-7 Properties of Exponents 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

15. Course 3 2-7 Properties of Exponents 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.

16. Course 3 2-7 Properties of Exponents THE ZERO POWER WordsNumbers Algebra The zero power of any number except 0 equals 1. 100 0 = 1 (–7) 0 = 1 a 0 = 1, if a  0

17. 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? Course 3 2-7 Properties of Exponents 10 18 - 5 A light-year is almost 10 13 km. 10 13 10 18 10 5 Subtract exponents. Additional Example 3: Astronomy Application

18. 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? Course 3 2-7 Properties of Exponents 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: Example 3

19. Course 3 2-7 Properties of Exponents Lesson Quiz: Part 1 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

20. Course 3 2-7 Properties of Exponents 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) Lesson Quiz: Part 2 Company A 6.