Lesson 13-7: Use Power of a Power and Product Properties

Property

If m, n, and x are real numbers, the Power of a Power property states:

This rule extends to products inside parentheses (Power of a Product property), meaning the outside power applies to every factor inside:

Examples

• Power of a Power: To simplify $(x^4)^3$, you multiply the exponents: $(x^4)^3 = x^{4 \cdot 3} = x^{12}$.
• Power of a Product: Apply the outside exponent to every factor inside the parentheses: $(a^2 b^5)^3 = (a^2)^3 (b^5)^3 = a^6 b^{15}$.
• With Negative Exponents: This rule works perfectly with negative exponents as well: $(y^{-3})^2 = y^{-3 \cdot 2} = y^{-6} = \frac{1}{y^6}$.

Explanation

Raising a power to another power is like making copies of copies! You have 'n' groups, and each group contains 'm' factors. To get the total number of factors, you simply multiply the two exponents together. It's the ultimate power-up move for your math skills, allowing you to bypass writing out huge strings of variables.

Property

When an expression requires multiple exponent rules, follow the order of operations.

First, use the Power Property $(a^m)^n = a^{m \cdot n}$ and the Product to a Power Property $(ab)^m = a^m b^m$ to clear any parentheses. Then, use the Product Property $a^m \cdot a^n = a^{m+n}$ to combine bases that are being multiplied.

Examples

• Example 1: To simplify $(x^2)^5 (x^4)^3$, first use the Power Property to clear the parentheses, getting $x^{10} \cdot x^{12}$. Then use the Product Property to add the exponents: $x^{10+12} = x^{22}$.
• Example 2: To simplify $(-3a^2b^3)^2$, distribute the exponent to every piece inside: $(-3)^2 (a^2)^2 (b^3)^2$. This simplifies to $9a^4b^6$.
• Example 3: To simplify $(2x^3)^2 (5x^4)$, first handle the power on the first term: $(4x^6)(5x^4)$. Now multiply the regular coefficients and add the exponents of the variables: $20x^{10}$.

Property

When evaluating complex expressions with multiple exponent properties, apply a systematic approach:

• simplify expressions within parentheses first,
• apply power of a power/product rules to remove parentheses,
• use product and quotient rules to combine identical bases, and
• finally rewrite the expression so it only contains positive exponents.

Examples

• Evaluate $\frac{(2x^3)^2 \cdot x^{-1}}{x^4}$:

First apply power of a product: $\frac{4x^6 \cdot x^{-1}}{x^4}$
Then product rule (numerator): $\frac{4x^5}{x^4}$
Finally quotient rule: $4x^1 = 4x$

• Simplify $(3a^{-2}b^4)^2 \cdot (ab)^{-3}$:

Apply power of a product to both terms: $9a^{-4}b^8 \cdot a^{-3}b^{-3}$
Combine like bases (add exponents): $9a^{-7}b^5$
Rewrite with positive exponents: $\frac{9b^5}{a^7}$

• Evaluate $\frac{(5^2)^{-1} \cdot 5^4}{5^0 \cdot 5^3}$:

Simplify powers and zero exponents: $\frac{5^{-2} \cdot 5^4}{1 \cdot 5^3} = \frac{5^2}{5^3} = 5^{-1} = \frac{1}{5}$