1. Exponents

2. An exponent tells how many times a number, or base is used as a factor Remember, Factor ∙ Factor = Product X 2 = X ∙ X where X is the base 2 is the exponent that tells how many times the base is multiplied by itself. Sometimes called the “power”

3. Exponents  The process of using exponents is called “raising to a power” where the exponent is the “power”.  The expression “5 4 ” is pronounced as “five, raised to the fourth power”  There are two specially named powers :  5 2 is five squared  5 3 is five cubed

4. Examples: X 5 = X ∙ X ∙ X ∙ X ∙ X4 2 = 4 ∙ 4 = 16 2 5 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 3211 3 = 11 ∙ 11 ∙ 11 = 1331 3 4 = 3 ∙ 3 ∙ 3 ∙ 3 = 8110 1 = 10 1.2 3 = 1.2 ∙ 1.2 ∙ 1.2 = 1.728.5 2 =.5 ∙.5 =.25 Try these:

5. 2-7 Properties of Exponents

6. 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

7. 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

8. 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.

9. 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.

10. 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.

11. 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

12. Multiply. Write the product as one power. With a coefficient, multiply those and keep the same base. E. 4X 2 2X 3 (4 2) (X 2 X 3 ) Multiply coefficient 8 X 2 + 3 Add exponents and keep 8X 5 the base the same F. 3X 5 8X 6 (3 8) (X 5 X 6 ) Multiply coefficient 24 X 5 + 6 Add exponents and keep 24X 11 the base the same

13. 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

14. 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

15. 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

16. 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

17. 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.

18. 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

19. Exponent Property Review

20. 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.

21. 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

22. Lesson Quiz 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

23. 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 Company A 6.