Lesson 40: Simplifying and Evaluating Expressions Using the Power Property of Exponents

New Concept

Power of a Power Property: If $m$ and $n$ are real numbers and $x \neq 0$, then $(x^m)^n = x^{mn}$.

What’s next

Next, you’ll apply this core idea to simplify products and quotients raised to a power, solving practical problems.

Property

If $m$ and $n$ are real numbers and $x \neq 0$, then $(x^m)^n = x^{mn}$.

Examples

• To simplify $(3^2)^4$, you multiply the exponents: $3^{2 \cdot 4} = 3^8$.
• For a variable, the rule is the same: $(x^5)^3 = x^{5 \cdot 3} = x^{15}$.
• This also works with coefficients: $((-2r)^2)^3 = (-2r)^{2 \cdot 3} = (-2r)^6 = 64r^6$.

Explanation

Think of this as stacking powers! When a power is raised to another power, you're just doing repeated multiplication. Instead of writing it all out, you can take a shortcut and simply multiply the two exponents together. This makes simplifying expressions with 'nested' exponents super quick and easy to handle.

Property

If $m$ is a real number with $x \neq 0$ and $y \neq 0$, then $(xy)^m = x^m y^m$.

Examples

• In $(3x^2y^5)^2$, the exponent 2 applies to everything: $3^2 \cdot (x^2)^2 \cdot (y^5)^2 = 9x^4y^{10}$.
• Be careful with negatives: $(-4a^3)^2 = (-4)^2 \cdot (a^3)^2 = 16a^6$.
• If a square garden has a side length of $5y$ feet, its area is $(5y)^2 = 25y^2$ square feet.

Explanation

Welcome to the 'share the power' party! The exponent outside the parentheses applies to every single factor inside. You have to distribute the power to each number and variable. Don't forget anyone, or your answer will be wrong! It ensures every part of the product gets raised to the same power.