How to Solve Powers of Products and Quotients? (+FREE Worksheet!)
When an entire product or quotient is raised to a power, each factor inside gets that exponent. These are the power of a product and the power of a quotient rules — two closely related properties that appear constantly in algebra. This lesson explains both rules, works through examples, and provides video lessons and practice.
What Are the Powers of Products and Quotients?
These two rules handle the situation where a parenthesis containing a product or fraction is raised to an exponent.
Power of a Product: (ab)n = an · bn
Power of a Quotient: (\(\color{blue}{\frac{a}{b}}\))n = an / bn (b ≠ 0)
Applying the Rules
Power of a Product
Distribute the outer exponent to every factor inside the parentheses.
- \(\color{blue}{(2x)^{3} = 2^{3} \cdot x^{3} = 8x^{3}}\)
- \(\color{blue}{(3y^{2})^{2} = 3^{2} \cdot (y^{2})^{2} = 9y^{4}}\)
Power of a Quotient
Apply the exponent to both the numerator and the denominator.
- \(\color{blue}{(\frac{x}{y})^{4} = \frac{x^{4}}{y^{4}}}\)
- \(\color{blue}{(\frac{2}{3})^{3} = \frac{2^{3}}{3^{3}} = \frac{8}{27}}\)
Combining both rules
When both a coefficient and a variable appear inside parentheses, distribute the exponent to each part:
- \(\color{blue}{(\frac{2x^{3}}{y^{2}})^{4} = 2^{4} \cdot \frac{x^{12}}{y^{8}} = \frac{16x^{12}}{y^{8}}}\)
Step-by-Step Summary
- Identify all factors inside the parentheses.
- Raise each factor to the outer exponent.
- For a product: multiply the resulting powers.
- For a quotient: the exponent goes to both numerator and denominator.
- Use the power-of-a-power rule (\(\color{blue}{(a^{m})^{n} = a^{\text{ mn }}}\)) when a base already has an exponent inside.
- Simplify coefficients.
Watch: Powers of Products and Quotients (Video Lesson)
Khan Academy demonstrates both rules with integer exponent examples step by step:
Powers of Products and Quotients – Worked Examples
Example 1: Simplify \(\color{blue}{(3x)^{2}}\).
Distribute the exponent: \(\color{blue}{3^{2} \cdot x^{2} = 9x^{2}}\).
Answer: \(\color{blue}{(3x)^{2} = 9x^{2}}\)
Example 2: Simplify \(\color{blue}{(2y^{3})^{4}}\).
Distribute: \(\color{blue}{2^{4} \cdot (y^{3})^{4} = 16 \cdot y^{12}}\).
Answer: \(\color{blue}{(2y^{3})^{4} = 16y^{12}}\)
Example 3: Simplify \(\color{blue}{(\frac{a}{b})^{5}}\).
Apply the power of a quotient rule: \(\color{blue}{\frac{a^{5}}{b^{5}}}\).
Answer: \(\color{blue}{(\frac{a}{b})^{5} = \frac{a^{5}}{b^{5}}}\)
Example 4: Simplify \(\color{blue}{(4x^{2})^{3}}\).
Distribute: \(\color{blue}{4^{3} \cdot (x^{2})^{3} = 64 \cdot x^{6}}\).
Answer: \(\color{blue}{(4x^{2})^{3} = 64x^{6}}\)
More Practice: Products and Exponents Video
This Khan Academy lesson covers products raised to exponents and the power-of-a-power property:
Exercises for Powers of Products and Quotients
Simplify each expression.
- \(\color{blue}{(3x)^{2}}\)
- \(\color{blue}{(2y^{3})^{4}}\)
- \(\color{blue}{(\frac{a}{b})^{5}}\)
- \(\color{blue}{(4x^{2})^{3}}\)
- \(\color{blue}{(\frac{x}{2})^{3}}\)
- \(\color{blue}{(5y)^{2}}\)
Answers
- \(\color{blue}{9x^{2}}\)
- \(\color{blue}{16y^{12}}\)
- \(\color{blue}{\frac{a^{5}}{b^{5}}}\)
- \(\color{blue}{64x^{6}}\)
- \(\color{blue}{\frac{x^{3}}{8}}\)
- \(\color{blue}{25y^{2}}\)
Free Powers of Products and Quotients Worksheet
Ready to practice on your own? Download our free Powers of Products and Quotients worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Powers of Products and Quotients before a quiz or test.
Download Properties of Exponents Worksheet
Frequently Asked Questions
What is the power of a product rule?
When a product is raised to a power, you raise each factor to that power: \(\color{blue}{(\text{ ab })^{n} = a^{n}b^{n}}\). For example, \(\color{blue}{(2x)^{4} = 16x^{4}}\).
What is the power of a quotient rule?
When a fraction is raised to a power, apply the exponent to both numerator and denominator: \(\color{blue}{(\frac{a}{b})^{n} = \frac{a^{n}}{b^{n}}}\). For example, \(\color{blue}{(\frac{x}{3})^{2} = \frac{x^{2}}{9}}\).
How does the power of a power rule fit in?
When a base already has an exponent and the whole thing is raised to another power, multiply the exponents: \(\color{blue}{(a^{m})^{n} = a^{\text{ mn }}}\). This rule often works together with the product and quotient power rules.
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