Laws of Exponents

5 2 = 5 x 5 = =3 x 3 x 3 x 3 = = 7 x 7 x 7 = 343

5 2 x 5 4 (5 x 5)(5 x 5 x 5 x 5) = 5 6 Do you see a pattern or shortcut?

3 3 x 3 5 (3x3x3)(3x3x3x3x3) = 3 8 Do you see a pattern or shortcut?

a 3 x a 5 (a x a x a)(a x a x a x a x a) = a 8 Do you see a pattern or shortcut?

Product of Powers Property To multiply powers (exponents) with the same base, add their exponents. a³ x a²= a = a 5

= 2 x 2 x 2 x 2 x 2 2 x 2 x 2 = = 4 x 4 x 4 x 4 x 4 4 x 4 = Do you see a pattern or shortcut?

Quotient of Powers Property To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. 6 8 = =

REVIEW When multiplying- add the exponents When dividing- subtract the exponents.

EXAMPLES = 69=69= 6464 z5z5 b b 7 b 3 = Z 8 = z 3

Zero Exponents For any nonzero number a, a 0 = 1 Anything to the zero power equals 1 (except zero) 4 0 = = 1 1

Negative Exponents For any nonzero number a and any integer n, a -n = 1/a n 5 -2 =

3 -5 = = y -2 =3 y2y2 a -7 b 3 =b3b3 a7a7

5 -8 x 5 -3 = = or a -2 x a 10 = a =a8a8

b -8 x b 5 = b =b -3 b3b3 1 or 3 -4 x 3 11 = =3737

= = 3 -3 or 1 3 a6a6 a -2 = a 6 – (-2) = a8a8 m2m2 m -4 = m 2 – (-4) = m6m6