The document discusses rules for operations involving exponents, including: - Multiplying powers with the same base by adding the exponents - Dividing powers with the same base by subtracting the exponents - Raising a power to a power by multiplying the exponents Examples are provided to illustrate multiplying, dividing, and raising powers to powers according to these rules. A quiz with problems applying the rules concludes the document.

1. Warm Up California Standards Lesson Presentation Preview

2. Warm Up 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

3. California Standards NS2.3 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base . NS2.1 Multiply, divide, and simplify rational numbers by using exponent rules.

4. The following suggests a rule for multiplying powers with the same base. 2 4 • 2 2 = (2 • 2 • 2 • 2) • (2 • 2) = 2 6 a 3 • a 2 = ( a • a • a ) • ( a • a ) = a 5 Notice that the sum of the exponents in each expression equals the exponent in the answer: 4 + 2 = 6 and 3 + 2 = 5.

6. Additional Example 1: Multiplying Powers with the Same Base B. n 5 • n 7 Add exponents. Add exponents. Simplify each expression. Write your answer in exponential form. A. 6 6 • 6 3 6 9 6 6 + 3 n 12 n 5 + 7

7. Check It Out! Example 1 B. x 2 • x 3 Add exponents. Add exponents. Simplify each expression. Write your answer in exponential form. A. 4 2 • 4 4 4 6 4 2 + 4 x 5 x 2 + 3

8. The following suggests a rule for dividing powers with the same base. Notice that the difference between the exponents in each expression equals the exponent in the answer: 6 – 2 = 4 and 5 – 3 = 2. 3 6 3 2 = = 3 • 3 • 3 • 3 = 3 4 3  3 3  3  3  3  3  3 1 1 1 1 x 5 x 3 = = x • x = x 2 x  x  x x  x  x  x  x 1 1 1 1 1 1

10. Subtract exponents. Additional Example 2: Dividing Powers with the Same Base Simplify each expression. Write your answer in exponential form. A. B. Subtract exponents. x 7 2 7 5 – 3 7 5 7 3 x 10 x 9 x 10 – 9 Think: x = x 1

11. Subtract exponents. 9 7 9 9 9 2 Check It Out! Example 2 A. B. e 10 e 5 Subtract exponents. Simplify each expression. Write your answer in exponential form. 9 9 – 2 e 10 – 5 e 5

12. RAISING A POWER TO A POWER To see what happens when you raise a power to a power, use the order of operations. ( c 3 ) 2 = ( c ● c ● c ) 2 = ( c ● c ● c ) ● ( c ● c ● c ) = c 6 Show the power inside the parentheses. Show the power outside the parentheses. Simplify.

13. RAISING A POWER TO A POWER Reading Math (9 4 ) 5 is read as “nine to the fourth power, to the fifth power.”

14. Simplify each expression. Write your answer in exponential form. Multiply exponents. Additional Example 3: 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.

15. Multiply exponents. Additional Example 3: Raising a Power to a Power C. D. (17 2 ) –20 17 2 • –20 17 –40 Simplify each expression. Write your answer in exponential form. 2 3 12 • – 3 2 3 – 36

16. Multiply exponents. Check It Out! Example 3 A. (3 3 ) 4 (3 3 ) 4 3 3 • 4 3 12 B. (4 8 ) 2 (4 8 ) 2 4 8 • 2 4 16 Multiply exponents. Simplify each expression. Write your answer in exponential form.

17. Multiply exponents. Check It Out! Example 3 C. D. (13 4 ) –10 13 4 • –10 13 –40 Simplify each expression. Write your answer in exponential form. 1 4 11• – 2 1 4 – 22

18. Lesson Quiz 3. 1. n 3  n 4 4. 5. 3 2 • 3 3 • 3 5 2. 8 • 8 8 6. ( m 2 ) 19 m 38 7. (9 -8 ) 9 8. (10 4 ) 0 1 Simplify each expression. Write your answer in exponential form. 8 9 n 7 10 9 10 5 10 4 t 2 3 10 t 9 t 7 1 9 72