Properties of Exponents – Part 2 Division and Zero Powers Learn zero exponents and division properties of exponents.

Division with the same base: Quotient of Powers Property Division with the same base: Keep base the same and subtract exponents am an = am-n Use when you are diving power a with another power that has the SAME BASE! 55 = 5(5-3) = 52 53

DIVIDING POWERS WITH THE SAME BASE Notice what occurs when you divide powers with the same base. 5 53 = 5  5  5 5  5  5  5  5 = 5 • 5 = 52 DIVIDING POWERS WITH THE SAME BASE Words Numbers Algebra To divide powers with the same base, keep the base and subtract the exponents. 6 5 9 – 4 9 4 = b m – n m n

Dividing Powers with the Same Base Divide. Write the quotient as one power. 7 5 3 x 10 9 1. 2. Subtract exponents Subtract 7 5 – 3 x 10 – 9 x 7 2 Think: x = x 1 e 10 9 9 4. 3. e 5 9 2 9 9 – 2 e 10 – 5 97 e 5

Dividing Powers with the Same Base Simplify 5. 32 x4 y4z Subtract exponents with like bases 6 x y2 9x(4-1)y(4-2) z Simplify fully 6 1.5x3y2 z

This result can be confirmed by writing out the factors. When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent. 1 = 4 2 42 – 2 = 40 = 1 = This result can be confirmed by writing out the factors. = (4 • 4) = 1 1 4 2

THE ZERO POWER Words Numbers Algebra The zero power of any number except 0 equals 1. 1000 = 1 (–7)0 = 1 a0 = 1, if a  0

0 does not exist because 00 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 0n .

Power of Quotient Property Fraction raised to a power a b m am bm = distribute power to each base in numerator and denominator and multiply it with existing exponent Use when you are raising an entire fraction to a power, distribute and multiply the exponents 3 2 = 22 32 = 4 9

Practice! 6 x 3 1. = x3 63 = x3 216 2. 4 2m = 24m4 44 = 16m4 256

Practice 1. 2. 3. 37 34 = 33 = 27 828 85 = 83 85 = 1 82 = 1 64 3 2 = 22 32 = 4 9

Try the Challenge Problem 16x3y ∙ -2xy -4xy3 -x Multiply straight across to make one fraction. -32x4y2 4x2y3 Divide whole numbers -8x4y2 x2y3 Subtract exponents and make sure all exponents in your answer are positive. -8x2 y