Section 5.1 The Rules of Exponents

Exponents An exponent is a “shorthand” number that saves writing the multiplication of the same numbers. exponent 34 base This is read “three to the fourth power.” 2

The Product Rule The Product Rule To multiply two exponential expressions that have the same base, keep the base and add the exponents. xa · xb = xa + b 3

Example Multiply. Numerical coefficient a. c4 · c5 = c4 + 5 = c9 b. 3a3 · a6 = 3a3 + 6 = 3a9 c. 4w2 · 2w5 = (4)(2)w2 + 5 = 8w7 4

Example Multiply.

The Quotient Rule The Quotient Rule (Part 1) Use this form if the larger exponent is in the numerator and x  0. 6

Example Divide. a. Note that the base does not change. b. 7

The Quotient Rule The Quotient Rule (Part 2) Use this form if the larger exponent is in the denominator and x  0. 8

Example Divide. a. Note that the base does not change. b. 9

Example Divide. a. b.

The Quotient Rule The Quotient Rule (Part 3) if x  0 (00 remains undefined). 11

Example Divide. a. b. 12

Raising a Power Raising a Power to a Power To raise a power to a power, keep the same base and multiply the exponents. 13

Example Simplify. a. (x5)3 = x5·3 = x15 b. (y3)3 = y3·3 = y9 14

Raising a Power Product Raised to a Power 15

Example Simplify. a. (2c)3 = (2)3c3 = 8c3 b. (5xy)2 = (5)2(xy)2 = 25x2y2 c. (4x3y2)3 = 43x9y6 = 64x9y6 16

Raising a Power Quotient Raised to a Power if y ≠ 0. 17

Example Simplify. a. b. 18

Example Simplify. Simplify inside the parenthesis first. Note that z0 = 1. Simplify the coefficient.