1. Properties of Exponents Learn to apply the properties of exponents and to evaluate the zero exponent.

2. 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

3. 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

4. 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 Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. Add exponents.

6. 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

7. Subtract exponents. 7 2 7 5 – 3 7 5 7 3 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

8. Subtract exponents. 9797 9 9 – 2 9 9 9 2 Try This: Divide. Write the quotient as one power. A. B. e 10 e 5 Subtract exponents. e 10 – 5 e 5

9. 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

10. 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! It is undefined! Helpful Hint 0n0n0n0n.

11. THE ZERO POWER WordsNumbers Algebra The zero power of any number except 0 equals 1. 100 0 = 1 (–7) 0 = 1 a = 1, if a  0

12. Practice Write the product or quotient as one power 1. n 3  n 4 2. 3. 4. 3 3 3 2 3 5 5. 8 8 8 = 10 9 10 5 t 9 t 7 1.n 7 2.10 4 3.t 2 4.3 10 5.8 9