1. 7.1 Properties of Exponents ©2001 by R. Villar All Rights Reserved

2. Properties of Exponents Consider the following… If x 3 means x x x and x 4 means x x x x then what is x 3 x 4 ? x x x x x x x x 7 Can you think of a quick way to come up with the solution?

3. Add the exponents. Your short cut is called the Product of Powers Property. Product of Powers Property: For all positive integers m and n: a m a n = a m + n Example: Simplify (3x 2 y 2 )(4x 4 y 3 ) Mentally rearrange the problem (using the commutative and associative properties). (3 4) (x 2 x 4 ) (y 2 y 3 ) 12x 6 y 5

4. Division Properties of Exponents How do you divide expressions with exponents? Examples:a 5 = a 3 x 3 = x 5 Let’s look at each problem in factored form. a a a a a a a a x x x x x x x x Notice that you can cancel from numerator to denominator. = a 2 = 1 x 2 Do you see a “short-cut” for dividing these expressions?

5. The short-cut is called the Quotient of Powers Property Quotient of Powers Property: a m = a m – n a n a ≠ 0 This means that when dividing with the same base, simply subtract the exponents. Examples:a 5 = a 3 a 5 – 3 = a 2

6. How do you raise a power to another power? Example: Simplify (x 2 ) 3 This means x 2 x 2 x 2 Using the Product of Powers Property gives x 6 What is the short-cut for getting from (x 2 ) 3 to x 6 ? Multiply the exponents. This short-cut is called the Power of a Power Property. Power of a Power Property: For all positive integers m and n: (a m ) n = a m n

7. Warm Up Simplify using one power. 1.(-3a ⁴b²)⁴ 2.(3ab³)(2a²b⁴) 3.-(5a³b⁴)² 4.18a⁴ 3a²

8. Example: Simplify (2m 3 n 5 ) 4 Raise each factor to the 4th power. (2 4 ) (m 3 ) 4 (n 5 ) 4 16m 12 n 20 The last problem was an example of how to use the Power of a Product Property. Power of a Product Property: For all positive integers m: (a b) m = a m b m

9. One final property is called the Quotient of Powers Property. This allows you to simplify expressions that are fractions with exponents. Quotient of Powers Property: Example: Evaluate This is the same as 3 3 4 3 = 27 64

10. Negative & Zero Exponents Study the table and think about the pattern. Exponent, n 5 4 3 2 1 0 –1 –2 –3 Power, 3 n Negative Exponents: a –n = 1 a n a cannot be zero 2438127931 1313 1919 1 27 What do you think 3 –4 will be? Zero Exponents: a 0 = 1 a cannot be zero 3 –4 = 1 = 1 3 4 81 This pattern suggests two definitions:

11. Example: Simplify 3y –3 x –2 This is the same as 3 1 x 2 1 y 3 1 = 3x 2 ` y 3 Example: Simplify 3 –8 3 5 Step down and add the exponents 3 –3 = 1 3 3 = 1 27 Example: Simplify (2x 4 ) –2 Step down and multiply the exponents 2 –2 x –8 = 1 1 2 2 x 8 = 1 4x 8

12. Remember, anything (other than 0) raised to the zero power is equal to 1 by definition. Example: (–8) 0 = 1 Example 5(–200x –6 y –2 z 20 ) 0 = 5(1) = 5