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.
In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex].
A General Note: 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
Example 7: Using the Power of a Product Rule
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
- [latex]{\left(a{b}^{2}\right)}^{3}[/latex]
- [latex]{\left(2t\right)}^{15}[/latex]
- [latex]{\left(-2{w}^{3}\right)}^{3}[/latex]
- [latex]\frac{1}{{\left(-7z\right)}^{4}}[/latex]
- [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex]
Solution
Use the product and quotient rules and the new definitions to simplify each expression.
- [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]
- [latex]2{t}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[/latex]
- [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]
- [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]
- [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]
Try It 7
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
a. [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex]
b. [latex]{\left(5t\right)}^{3}[/latex]
c. [latex]{\left(-3{y}^{5}\right)}^{3}[/latex]
d. [latex]\frac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex]
e. [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex]
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.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.
A General Note: 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
Example 8: Using the Power of a Quotient Rule
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
- [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}[/latex]
- [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}[/latex]
- [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}[/latex]
- [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex]
- [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex]
Solution
- [latex]{\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]
- [latex]{\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]
- [latex]{\\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]
- [latex]{\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]
- [latex]{\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]
Try It 8
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
a. [latex]{\left(\frac{{b}^{5}}{c}\right)}^{3}[/latex]
b. [latex]{\left(\frac{5}{{u}^{8}}\right)}^{4}[/latex]
c. [latex]{\left(\frac{-1}{{w}^{3}}\right)}^{35}[/latex]
d. [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex]
e. [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex]