4.1 Properties of Exponents 1/28/2013

Power, Base and Exponent: 72

In general: am•an = am+n Product of Powers: Ex. 32 • 35 = 3•3•3•3•3•3•3 = 37 = 32+5 In general: am•an = am+n To multiply powers with the same base, keep the base and add the exponents.

In general: (am)n = am•n Power of a Power: Ex. (23 )2 = (23 )• (23 ) =(2•2•2)•(2•2•2) = 26 = 23•2 In general: (am)n = am•n To raise a power to a power, keep the base and multiply the exponents.

In general: (a •b)m =a m•b m Power of a Product: Ex. (4•3 )3 = (4•3 )• (4•3 ) • (4•3 ) =(4•4•4)•(3•3•3) = 4 3 •3 3 In general: (a •b)m =a m•b m

In general: a0 = 1 Zero Exponent 50 = 1 40 = 1 30 = 1 ÷5 ÷5 ÷5 ÷4 ÷4 ÷3 ÷3 ÷3 In general: a0 = 1 Any base raised to a 0 power equals 1.

Exponential Form Fraction Form 33 27 32 9 31 3 30 1 3-1 1 3 3-2 1 9 = 1 3 2 3-3 1 27 = 1 3 3

Ex. 5-2 Ex. In general: Negative Exponent Negative exponent MOVES power. If the power with a negative exponent is in the numerator, the power moves to the denominator and exponent becomes positive. If the power with a negative exponent is in the denominator, the power moves to the numerator and exponent becomes positive.

Ex. In general: Quotient of Powers To divide powers with the same base, keep the base and subtract the exponents.

Power of a Quotient Ex. In general:

Example 1 ( ) 8 2 – )4 = ( ) 8 4 2 – + = ( ) 4 2 – 1 ( )4 2 – = = 16 1 Evaluate Expressions with Negative Exponents ( ) 8 2 – )4 Product of powers property = ( ) 8 4 2 – + = ( ) 4 2 – Simplify exponent. 1 ( )4 2 – = Negative exponent property = 16 1 Evaluate power.

Example 2 33 35 2 Evaluate . 33 35 2 = ( )2 32 = 34 = 81 Evaluate Quotients with Exponents 33 35 2 Evaluate . Quotient of powers property 33 35 2 = ( )2 32 = 34 Power of a power property = 81 Evaluate power. 12

Evaluate the expression. Checkpoint Evaluate Numerical Expressions Evaluate the expression. 1. ( )3 22 ANSWER 64 ( )3 50 2. ANSWER 1 ANSWER 27 1 – 3. ( ) 5 3 – )2 4. 3 2 ANSWER 27 8

Example 3 a. y x – 3 2 x 2 ( )2 y – 3 = = x 2 y – 3 2 • = x 2 y – 6 = Simplify Algebraic Expressions a. y x – 3 2 Power of a quotient property x 2 ( )2 y – 3 = = x 2 y – 3 2 • Power of a power property = x 2 y – 6 Simplify exponent. = x 2y 6 Negative exponent property

Example 3 ( )2 5y – 3 y 5y b. = ( )2 y – 3 y 5y 52 = 25y y 5y – 3 2 • Simplify Algebraic Expressions ( )2 5y – 3 y 5y b. = ( )2 y – 3 y 5y 52 Power of a product property = 25y y 5y – 3 2 • Power of a power property = 25y y 5y – 6 Simplify exponent. = 25y – 6 5 1 + Product of powers property = 25y 0 Simplify exponent. = 25 Zero exponent property

Example 3 c. x 5y 2 – x 3y 6 = x – 3 5 y 6 ( 2) = x 2y 8 – = y 8 x 2 Simplify Algebraic Expressions c. x 5y 2 – x 3y 6 = x – 3 5 y 6 ( 2) Quotient of powers property = x 2y 8 – Simplify exponent. = y 8 x 2 Negative exponent property 16

Homework: WS 4.1 odd problems