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Study Guides > Intermediate Algebra

Read: Find the Power of a Product and a Quotient

Learning Objectives

  • Simplify an expression with a product raised to a power
  • Simplify an expression with a quotient raised to a power
  • Combine all the exponents rules to simplify expressions
 

Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider [latex]{\left(pq\right)}^{3}[/latex]. We begin by using the associative and commutative properties of multiplication to regroup the factors.
[latex]\begin{array}{ccc}\hfill {\left(pq\right)}^{3}& =& \stackrel{3\text{ factors}}{{\left(pq\right)\cdot \left(pq\right)\cdot \left(pq\right)}}\hfill \\ & =& p\cdot q\cdot p\cdot q\cdot p\cdot q\hfill \\ & =& \stackrel{3\text{ factors}}{{p\cdot p\cdot p}}\cdot \stackrel{3\text{ factors}}{{q\cdot q\cdot q}}\hfill \\ & =& {p}^{3}\cdot {q}^{3}\hfill \end{array}[/latex]
In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex].

The Power of a Product Rule of Exponents

For any real numbers [latex]a[/latex] and [latex]b[/latex] and any integer [latex]n[/latex], the power of a product rule of exponents states that
[latex]{\left(ab\right)}^{n}={a}^{n}{b}^{n}[/latex]
 

Example

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
  1. [latex]{\left(a{b}^{2}\right)}^{3}[/latex]
  2. [latex]{\left(2^a{t}\right)}^{15}[/latex]
  3. [latex]{\left(-2{w}^{3}\right)}^{3}[/latex]
  4. [latex]\frac{1}{{\left(-7z\right)}^{4}}[/latex]
  5. [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex]

Answer: Use the product and quotient rules and the new definitions to simplify each expression.

  1. [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex]
  2. [latex]{\left(2^a{t}\right)}^{15}={\left(2^a\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{a\cdot15}\cdot{t}^{15}=2^{15a}\cdot{t}^{15}[/latex]
  3. [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex]
  4. [latex]\frac{1}{{\left(-7z\right)}^{4}}=\frac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\frac{1}{2,401{z}^{4}}[/latex]
  5. [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\frac{{f}^{14}}{{e}^{14}}[/latex]

We may even encounter In the following video,  we provide more examples of how to find the power of a product. https://youtu.be/p-2UkpJQWpo

Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
[latex]{\left({e}^{-2}{f}^{2}\right)}^{7}=\frac{{f}^{14}}{{e}^{14}}[/latex]
Let’s rewrite the original problem differently and look at the result.
[latex]\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex]
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.
[latex]\normalsize\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{\left({f}^{2}\right)}^{7}}{{\left({e}^{2}\right)}^{7}}\hfill \\ & =& \frac{{f}^{2\cdot 7}}{{e}^{2\cdot 7}}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex]

The Power of a Quotient Rule of Exponents

For any real numbers [latex]a[/latex] and [latex]b[/latex] and any integer [latex]n[/latex], the power of a quotient rule of exponents states that
[latex]{\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}[/latex]

Example

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
  1. [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}[/latex]
  2. [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}[/latex]
  3. [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}[/latex]
  4. [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex]
  5. [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex]

Answer:

  1. [latex]\Large{\left(\frac{4}{{z}^{11}}\right)}^{3}=\frac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\frac{64}{{z}^{11\cdot 3}}=\frac{64}{{z}^{33}}[/latex]
  2. [latex]\Large{\left(\frac{p}{{q}^{3}}\right)}^{6}=\frac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\frac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\frac{{p}^{6}}{{q}^{18}}[/latex]
  3. [latex]\Large{\left(\frac{-1}{{t}^{2}}\right)}^{27}=\frac{{\left(-1\right)}^{27}}{{\left({t}^{2}\right)}^{27}}=\frac{-1}{{t}^{2\cdot 27}}=\frac{-1}{{t}^{54}}=-\frac{1}{{t}^{54}}[/latex]
  4. [latex]\Large{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\frac{{j}^{3}}{{k}^{2}}\right)}^{4}=\frac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\frac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\frac{{j}^{12}}{{k}^{8}}[/latex]
  5. [latex]\Large{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\frac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\frac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\frac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\frac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\frac{1}{{m}^{6}{n}^{6}}[/latex]

The following video provides more examples of simplifying expressions using the power of a quotient and other exponent rules. https://youtu.be/BoBe31pRxFM

Summary

  • Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.
  • The product rule for exponents: For any number x and any integers a and b, [latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex].
  • The quotient rule for exponents: For any non-zero number x and any integers a and b: [latex] \displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex]
  • The power rule for exponents:
    1. For any nonzero numbers a and b and any integer x, [latex]\left(ab\right)^{x}=a^{x}\cdot{b^{x}}[/latex].
    2. For any number a, any non-zero number b, and any integer x, [latex] \displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex]

Licenses & Attributions

CC licensed content, Original

  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
  • Simplify Expressions Using Exponent Rules (Power of a Product). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Simplify Expressions Using Exponent Rules (Power of a Quotient). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

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