Lesson 13-6: Use Product and Quotient of Powers Properties

Property

When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$.
When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$.

Examples

• Product Property: Multiply $10^4 \cdot 10^2$.

$10^4 \cdot 10^2 = 10^{4+2} = 10^6$.

• Quotient Property: Divide $\frac{10^7}{10^3}$.

$\frac{10^7}{10^3} = 10^{7-3} = 10^4$.

• Using Negative Exponents: Multiply $x^{-5} \cdot x^2$.

$x^{-5} \cdot x^2 = x^{-5+2} = x^{-3}$, which can be written as $\frac{1}{x^3}$.

Explanation

These exponent properties act as mathematical shortcuts because multiplication is just repeated addition, and division is repeated subtraction. When you multiply powers of the same base, you are combining groups of factors, so you add the exponents. When you divide, you are canceling out groups of factors, so you subtract the exponents.

Property

The Product and Quotient of Powers properties only apply to powers with the exact same base. Exponents cannot be added or subtracted when the bases differ.

$a^m \cdot b^n$ cannot be simplified further.

Examples

• Correct Simplification: Simplify $2^3 \cdot 3^4 \cdot 2^5$.

Group the identical bases together to add their exponents: $2^{3+5} \cdot 3^4 = 2^8 \cdot 3^4$.

• Error Analysis: A student simplified $2^3 \cdot 5^4$ as $10^7$.

This is incorrect because the bases (2 and 5) are different, so their exponents cannot be combined. The expression $2^3 \cdot 5^4$ is already fully simplified.

• Algebraic Expression: Simplify $\frac{x^6 \cdot y^4}{x^2 \cdot y^3}$.

Apply the quotient property to each base separately: $x^{6-2} \cdot y^{4-3} = x^4 \cdot y^1 = x^4y$.

Explanation

When simplifying expressions with multiple variables or numbers, you must act like a sorter: group identical bases together and apply the exponent rules to each group separately. A common mistake is trying to add exponents across different bases, such as treating $2^3 \cdot 3^2$ as $6^5$. Always leave powers with different bases separate in your final simplified answer.

Property

When expressions contain both products and quotients of powers with the same base, apply the product property first to simplify the numerator and denominator separately, and then apply the quotient property:

Examples

• Example 1: $\frac{x^5 \cdot x^3}{x^2} = \frac{x^{5+3}}{x^2} = \frac{x^8}{x^2} = x^{8-2} = x^6$
• Example 2: $\frac{y^4}{y^2 \cdot y^5} = \frac{y^4}{y^{2+5}} = \frac{y^4}{y^7} = y^{4-7} = y^{-3} = \frac{1}{y^3}$
• Example 3: $\frac{a^7 \cdot a^2}{a^3 \cdot a^4} = \frac{a^{7+2}}{a^{3+4}} = \frac{a^9}{a^7} = a^{9-7} = a^2$

Explanation

When working with complex algebra fractions involving both multiplication and division of powers, follow a systematic two-step approach. First, clean up the top and bottom: use the product property to combine powers in the numerator and denominator separately by adding their exponents. Once you have a single power on top and a single power on the bottom, apply the quotient property by subtracting the exponents.