Multiplication Properties of Exponents Section 10.2

Product Rule for Exponents In other words, to multiply two exponential expressions with the same base, keep the base and add the exponents. This is called simplifying the exponential expression. am  an  a m   n n n n If m and n are positive integers and a is a real number, then For example, 2

Helpful Hint Usually, an exponent of 1 is not written, so when no exponent appears, we assume that the exponent is 1. For example, 2 = 2 1 and 7 = 7 1. Martin-Gay, Prealgebra, 5ed 3

Helpful Hint These examples will remind you of the difference between adding and multiplying terms. AdditionMultiplication 7x 2  5x 2  7x 2  5x 2  4x  5x 3  4x  5x 3  12x 2 35x 4 20x 4 4x  5x 3 Terms cannot be combined By the product rule exponents were added. By the distributive property terms were added. 4

Power Rule for Exponents In other words, to raise an exponential expression to a power, keep the base and multiply the exponents. (am)n  am n If m and n are positive integers and a is a real number, then For example, Martin-Gay, Prealgebra, 5ed 5

Helpful Hint Take a moment to make sure that you understand when to apply the product rule and when to apply the power rule. Product Rule  Power Rule  Add Exponents Multiply Exponents 6

Power of a Product Rule In other words, to raise a product to a power, raise each factor to the power. (ab)n  a nb n If n is a positive integer and a and b are real numbers, then For example, Martin-Gay, Prealgebra, 5ed 7