Negative and Zero Exponent Rules
Learning Outcomes
- Simplify exponential expressions containing negative exponents and exponents of 0 and 1
Define and Use the Zero Exponent Rule
Return to the quotient rule. We worked with expressions for which [latex]a>b[/latex] so that the difference [latex]a-b[/latex] would never be zero or negative.The Quotient (Division) 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][latex]\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex]
If we were to simplify the original expression using the quotient rule, we would have:The Zero Exponent Rule of Exponents
For any nonzero real number [latex]a[/latex], the zero exponent rule of exponents states thatExample
Simplify each expression using the zero exponent rule of exponents.- [latex]\dfrac{{c}^{3}}{{c}^{3}}[/latex]
- [latex]\dfrac{-3{x}^{5}}{{x}^{5}}[/latex]
- [latex]\dfrac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex]
- [latex]\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex]
Answer: Use the zero exponent and other rules to simplify each expression. 1. [latex-display]\Large\begin{array}\text{ }\frac{c^{3}}{c^{3}} \hfill& =c^{3-3} \\ \hfill& =c^{0} \\ \hfill& =1\end{array}[/latex-display] 2. [latex-display]\Large\begin{array}{ccc}\hfill \frac{-3{x}^{5}}{{x}^{5}}& =& -3\cdot \frac{{x}^{5}}{{x}^{5}}\hfill \\ & =& -3\cdot {x}^{5 - 5}\hfill \\ & =& -3\cdot {x}^{0}\hfill \\ & =& -3\cdot 1\hfill \\ & =& -3\hfill \end{array}[/latex-display] 3. [latex-display]\large\begin{array}{cccc}\hfill \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}}\hfill &\text{Use the product rule in the denominator}.\hfill \\ & =&\frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}}\hfill & \text{Simplify}.\hfill \\ & =&{\left({j}^{2}k\right)}^{4 - 4}\hfill &\text{Use the quotient rule}.\hfill \\ & =& {\left({j}^{2}k\right)}^{0}\hfill&\text{Simplify}.\hfill \\ & =& 1& \end{array}[/latex-display] 4. [latex]\large\begin{array}{cccc}\hfill \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& =&5{\left(r{s}^{2}\right)}^{2 - 2}\hfill &\text{Use the quotient rule}.\hfill \\ & =& 5{\left(r{s}^{2}\right)}^{0}\hfill &\text{Simplify}.\hfill \\ & =& 5\cdot 1\hfill &\text{Use the zero exponent rule}.\hfill \\ & =& 5\hfill &\text{Simplify}.\hfill \end{array}[/latex]
Define and Use the Negative Exponent Rule
Another useful result occurs if we relax the condition that [latex]a>b[/latex] in the quotient rule even further. For example, can we simplify [latex]\frac{{h}^{3}}{{h}^{5}}[/latex]? When [latex]a<b[/latex]—that is, where the difference [latex]a-b[/latex] is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal. Divide one exponential expression by another with a larger exponent. Use our example, [latex]\frac{{h}^{3}}{{h}^{5}}[/latex].The Negative Rule of Exponents
For any nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states thatExample
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.- [latex]\dfrac{{(2b) }^{3}}{{(2b) }^{10}}[/latex]
- [latex]\dfrac{{z}^{2}\cdot z}{{z}^{4}}[/latex]
- [latex]\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex]
Answer:
- [latex]\dfrac{{(2b) }^{3}}{{(2b)}^{10}}={(2b)}^{3 - 10}={(2b) }^{-7}=\frac{1}{{(2b)}^{7}}[/latex]
- [latex]\dfrac{{z}^{2}\cdot z}{{z}^{4}}=\frac{{z}^{2+1}}{{z}^{4}}=\frac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\frac{1}{z}[/latex]
- [latex]\dfrac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\frac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex]
Combine Exponent Rules to Simplify Expressions
Now we will combine the use of the product and quotient rules to simplify expressions whose terms may have negative or zero exponents.Example
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.- [latex]{b}^{2}\cdot {b}^{-8}[/latex]
- [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex]
- [latex]\frac{-7z}{{\left(-7z\right)}^{5}}[/latex]
Answer:
- [latex]{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex]
- [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1[/latex]
- [latex]\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}[/latex]
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Simplify Expressions Using the Quotient and Zero Exponent Rules. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Expressions Using the Quotient and Negative Exponent Rules Mathispower4u . Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Expressions Using Exponent Rules (Product, Quotient, Zero Exponent). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: Lumen Learning Authored by: Jay Abrams et, al.. License: CC BY: Attribution.
- Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.