1. Pre-Algebra 2-7 Properties of Exponents Properties of Exponents 8-3/8-4/8-5 Pre-Algebra Multiply Powers Multiply Powers Dividing Powers Dividing Powers Raising a Power to a Power Raising a Power to a Power

2. Pre-Algebra 2-7 Properties of Exponents 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

3. Pre-Algebra 2-7 Properties of Exponents Words NumbersAlgebra To raise a power to a power, multiply the exponents. (b m ) n = b mn (3 5 ) 2 = 3 10 = 59,049 RAISING A POWER TO A POWER

4. Pre-Algebra 2-7 Properties of Exponents Additional Example 1A & 1B: 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. Pre-Algebra 2-7 Properties of Exponents D. 24 4 24 4 C. 2 5 2 2 6 2 5 + 1 24 8 4 + 4 Think: 2 = 2 1 Additional Example 1: Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. Add exponents.

6. Pre-Algebra 2-7 Properties of Exponents Try This: Example 1A & 1B A. ( 4 2 ) 4 65,536 4 8 B. ( x 2 ) 3 x 6 Simplify.

7. Pre-Algebra 2-7 Properties of Exponents D. ( 41 2 ) 7 C. x 5 y 2 41 14 Try This: Example 1C & 1D Multiply. Write the product as one power. Cannot combine; the bases are not the same. x 5 y 2

8. Pre-Algebra 2-7 Properties of Exponents 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

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

10. Pre-Algebra 2-7 Properties of Exponents Subtract exponents. 9797 9 9 – 2 9 9 9 2 Try This: Example 2 Divide. Write the product as one power. A. B. e 10 e 5 Subtract exponents. e 10 – 5 e 5

11. Pre-Algebra 2-7 Properties of Exponents 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

12. Pre-Algebra 2-7 Properties of Exponents 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. Additional Example 3: Astronomy Application

13. Pre-Algebra 2-7 Properties of Exponents 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 Try This: Example 3

14. Pre-Algebra 2-7 Properties of Exponents Lesson Quiz: Part 1 Write the product or quotient as one power 3. 8 9 n 12 1. (n 3 ) 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