2-7

1. 2-7 Properties of Exponents Warm Up Problem of the Day Lesson Presentation Pre-Algebra

2. 2-7 Properties of Exponents Pre-Algebra Warm Up Evaluate. 27 1. 33 2. 4 • 4 • 4 • 4 256 3.b2 for b = 4 16 4. n2r for n = 3 and r = 2 18

3. Problem of the Day Calculate 6 to the fourth power minus 56. 1240

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

5. The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 • 7 • 7 • 7 = 74 (7 • 7 • 7) • 7 = 73 • 71 = 74 (7 • 7) • (7 • 7) = 72 • 72 = 74

6. MULTIPLYING POWERS WITH THE SAME BASE Numbers Algebra Words To multiply powers with the same base, keep the base and add the exponents. bm • bn = bm+ n 35• 38 = 35 + 8 = 313

7. A. 66 •63 6 6 + 3 6 9 n 5 + 7 n 12 Additional Example 1A & 1B: Multiplying Powers with the Same Base Multiply. Write the product as one power. Add exponents. B. n5 •n7 Add exponents.

8. 1 Think: 2 = 2 2 5 + 1 2 6 24 4 + 4 24 8 Additional Example 1: Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. C. 25 • 2 Add exponents. D.244• 244 Add exponents.

9. A. 42 •44 4 2 + 4 4 6 x 2 + 3 x 5 Try This: Example 1A & 1B Multiply. Write the product as one power. Add exponents. B. x2 •x3 Add exponents.

10. 41 2 + 7 41 9 Try This: Example 1C & 1D Multiply. Write the product as one power. C. x5 • y2 Cannot combine; the bases are not the same. x5 • y2 D.412• 417 Add exponents.

11. 5555 5 5555 5 5 5 = = = 5 •5 = 52 555 555 53 DIVIDING POWERS WITH THE SAME BASE Words Numbers Algebra To divide powers with the same base, keep the base and subtract the exponents. 6 b 9 m = = = 6 b 6 9 – 4 m – n 5 6 b 4 n Notice what occurs when you divide powers with the same base.

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

13. 9 – 2 9 10 – 5 e 5 e Try This: Example 2 Divide. Write the product as one power. 9 9 A. 9 2 Subtract exponents. 97 e 10 B. e 5 Subtract exponents.

14. 4 2 = 42 – 2 = 40 = 1 1 = 2 4 1 1 (4 • 4) (4 • 4) 4 2 = 1 = = = (4 • 4) (4 • 4) 4 2 When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0exponent. This result can be confirmed by writing out the factors.

15. Helpful Hint 0 does not exist because 00 represents a quotient of the form But the denominator of this quotient is 0, which is impossible, since you cannot divide by 0. . 0n 0n 0

16. THE ZERO POWER Algebra Words Numbers The zero power of any number except 0 equals 1. 1000 = 1 a0= 1, if a 0 (–7)0 = 1

17. 18 10 5 10 18 - 5 10 13 10 A light-year is almost 1013 km. Additional Example 3: Astronomy Application A light-year, or the distance light travels in one year, is almost 1018 centimeters. To convert this number to kilometers, you must divide by 105. How many kilometers is a light-year? Subtract exponents.

18. 10 7 3 10 The weight in metric tons is equal to the weight in kilograms divided by 10 kilograms per metric ton. 7 - 3 10 3 4 10 The ship had 104 metric tons of grain loaded. Try This: Example 3 A ship has 107 kilograms of grain loaded into its cargo hold. A metric ton is 103 kilograms. How many metric tons of grain were loaded? Subtract exponents.

19. n 7 8 9 109 4 10 105 t 2 t9 t7 3 10 Lesson Quiz: Part 1 Write the product or quotient as one power 1. n3n4 2. 8 • 88 3. 4. 5. 33 • 32 • 35

20. Lesson Quiz: Part 2 6. 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 = 122 items) Company A