1. Properties of Exponents

2. If a number is in exponential form, the exponent represents how many times the base is to be used as a factor. A number produced by raising a base to an exponent is called a power. Both 27 and 3 3 represent the same power. 7 Exponent Base 2 { Power

3. Product Rule: Same base, add exponents. Why does the rule work?

4. Example: Multiplying Powers with the Same Base 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.

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

6. E. 4 2 4 4 4 6 4 2 + 4 F. x 2 x 3 x 5 x 2 + 3 Add exponents. Multiply. Write the product as one power.

7. H. 41 2 41 7 G. 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

8. Quotient Rule: Same base, subtract exponents. Why does the rule work? Notice what occurs when you divide powers with the same base. 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

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

10. Subtract exponents. 9797 9 9 – 2 9 9 9 2 Divide. Write the product as one power. C. D. e 10 e 5 Subtract exponents. e 10 – 5 e 5

11. Reading Math (9 ) is read as “nine to the fourth, to the fifth.” 5 4 Power Rule: Power raised to a power, multiply exponents.

12. Simplify. Multiply exponents. Example: Raising a Power to a Power A. (5 4 ) 2 (5 4 ) 2 = 5 4 2 = 5 8 B. (6 7 ) 9 (6 7 ) 9 = 6 7 9 = 6 63 Multiply exponents.

13. Simplify. Multiply exponents. C. (3 3 ) 4 (3 3 ) 4 = 3 3 4 = 3 12 D. (4 8 ) 2 (4 8 ) 2 = 4 8 2 = 4 16 Multiply exponents. Example: Raising a Power to a Power

14. Simplify. Multiply exponents. Example: Raising a Power to a Power E. F. (17 2 ) 20 (17 2 ) 20 = 17 2 20 = 17 40 Multiply exponents. 2323 123 2323 3 = 2323 36 =

15. Simplify. Multiply exponents. Example G. H. (13 4 ) 10 (13 4 ) 10 = 13 4 10 = 13 40 Multiply exponents. 1414 112 1414 2 = 1414 22 =

16. Power of Product Rule Product raised to a power, distribute power to each base. Evaluate the number(s)/coefficients. Multiply the exponents. (2x 2 y 5 ) 3 = 2 3 x 2*3 y 5 * 3 8x 6 y 15

17. A. (4a 3 b 4 ) 2 = 16a 6 b 8 B. (9x 5 y 4 ) 3 = 729x 15 y 12 Example: Raising a Product to a Power

18. C. (5a 5 b 10 ) 3 = 125a 15 b 30 D. (4x 7 y 9 ) 4 = 256x 28 y 36 Example: Raising a Product to a Power

19. Lesson Quiz Write the product or quotient as one power. 8 9 1. n 3  n 4 10 9 10 5 10 4 4. t 2 5. 3 2 3 3 3 5 3 10 2. 8 8 8 t 9 t 7 6. (m 2 ) 19 m 38 n 7