Zero And Negative Exponents | College Algebra - Lumen Learning

Learning Outcomes

• Simplify expressions with exponents equal to zero.
• Simplify expressions with negative exponents.
• Simplify exponential expressions.

Return to the quotient rule. We made the condition that [latex]m>n[/latex] so that the difference [latex]m-n[/latex] would never be zero or negative. What would happen if [latex]m=n[/latex]? In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.

[latex]\dfrac{t^{8}}{t^{8}}=\dfrac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex]

If we were to simplify the original expression using the quotient rule, we would have

[latex]\dfrac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex]

If we equate the two answers, the result is [latex]{t}^{0}=1[/latex]. This is true for any nonzero real number, or any variable representing a real number.

[latex]{a}^{0}=1[/latex]

The sole exception is the expression [latex]{0}^{0}[/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.

A General Note: The Zero Exponent Rule of Exponents

For any nonzero real number [latex]a[/latex], the zero exponent rule of exponents states that

[latex]{a}^{0}=1[/latex]

Example: Using the Zero Exponent Rule

Simplify each expression using the zero exponent rule of exponents.

1. [latex]\dfrac{{c}^{3}}{{c}^{3}}[/latex]
2. [latex]\dfrac{-3{x}^{5}}{{x}^{5}}[/latex]
3. [latex]\dfrac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex]
4. [latex]\dfrac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex]

Show Solution

Use the zero exponent and other rules to simplify each expression.

1. [latex]\begin{align}\frac{c^{3}}{c^{3}} & =c^{3-3} \\ & =c^{0} \\ & =1\end{align}[/latex]
2. [latex]\begin{align} \frac{-3{x}^{5}}{{x}^{5}}& = -3\cdot \frac{{x}^{5}}{{x}^{5}} \\ & = -3\cdot {x}^{5 - 5} \\ & = -3\cdot {x}^{0} \\ & = -3\cdot 1 \\ & = -3 \end{align}[/latex]
3. [latex]\begin{align} \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}} && \text{Use the product rule in the denominator}. \\ & = \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}} && \text{Simplify}. \\ & = {\left({j}^{2}k\right)}^{4 - 4} && \text{Use the quotient rule}. \\ & = {\left({j}^{2}k\right)}^{0} && \text{Simplify}. \\ & = 1 \end{align}[/latex]
4. [latex]\begin{align} \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& = 5{\left(r{s}^{2}\right)}^{2 - 2} && \text{Use the quotient rule}. \\ & = 5{\left(r{s}^{2}\right)}^{0} && \text{Simplify}. \\ & = 5\cdot 1 && \text{Use the zero exponent rule}. \\ & = 5 && \text{Simplify}. \end{align}[/latex]

Try It

Simplify each expression using the zero exponent rule of exponents.

1. [latex]\dfrac{{t}^{7}}{{t}^{7}}[/latex]
2. [latex]\dfrac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex]
3. [latex]\dfrac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex]
4. [latex]\dfrac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex]

Show Solution

1. [latex]1[/latex]
2. [latex]\dfrac{1}{2}[/latex]
3. [latex]1[/latex]
4. [latex]1[/latex]

In this video we show more examples of how to simplify expressions with zero exponents.

Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that [latex]m>n[/latex] in the quotient rule even further. For example, can we simplify [latex]\dfrac{{h}^{3}}{{h}^{5}}[/latex]? When [latex]m