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. 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

5. 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

6. 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?

7. 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 x 3 = x 5 a 5 – 3 = a 2 x 3 – 5 = x –2 = 1 x 2

8. 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

9. Stairway to the Exponents mult/div add/subt move down a step Mult/Divide Add/Subtract Power Here’s a tool you may want to use to help you remember the properties for exponents. The steps represent the Order of Operations. When working with exponents, step down to the next lower step. For example, when multiplying expressions with exponents, step down and add the exponents.

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