Simplifying Expressions with Negative Exponents and Exponents of 0 and 1
Learning Outcomes
- Simplify exponential expressions containing negative exponents
- Simplify exponential expressions containing exponents of 0 and 1
Quotient Property of Exponents
If [latex]a[/latex] is a real number, [latex]a\ne 0[/latex], and [latex]m,n[/latex] are whole numbers, then [latex-display]{\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n\text{ and }{\Large\frac{{a}^{m}}{{a}^{n}}}={\Large\frac{1}{{a}^{n-m}}},n>m[/latex-display]Define and use the negative exponent rule
We now propose another question about exponents. Given a quotient like [latex] \displaystyle \frac{{{2}^{m}}}{{{2}^{n}}}[/latex] what happens when n is larger than m? We will need to use the negative rule of exponents to simplify the expression so that it is easier to understand. Let's look at an example to clarify this idea. Given the expression:[latex]\frac{{h}^{3}}{{h}^{5}}[/latex]
Expand the numerator and denominator, all the terms in the numerator will cancel to [latex]1[/latex], leaving two hs multiplied in the denominator, and a numerator of [latex]1[/latex].[latex]\begin{array}{r}\frac{h^{3}}{h^{5}}\,\,\,=\,\,\,h^{3-5}\\\\=\,\,\,h^{-2}\,\,\end{array}[/latex]
Putting the answers together, we have [latex]{h}^{-2}=\frac{1}{{h}^{2}}[/latex]. This is true when h, or any variable, is a real number and is not zero.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
Evaluate the expression [latex]{4}^{-3}[/latex].
Answer: First, write the expression with positive exponents by putting the term with the negative exponent in the denominator.
[latex]{4}^{-3} = \frac{1}{{4}^{3}} = \frac{1}{4\cdot4\cdot4}[/latex]
Now that we have an expression that looks somewhat familiar.[latex]\frac{1}{4\cdot4\cdot4} = \frac{1}{64}[/latex]
Answer
[latex-display]\frac{1}{64}[/latex-display]example
Simplify: 1. [latex]{4}^{-2}[/latex] 2. [latex]{x}^{-3}[/latex] Solution1. | |
[latex]{4}^{-2}[/latex] | |
Use the definition of a negative exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex]. | [latex]{\Large\frac{1}{{4}^{2}}}[/latex] |
Simplify. | [latex]{\Large\frac{1}{16}}[/latex] |
2. | |
[latex]{x}^{-3}[/latex] | |
Use the definition of a negative exponent, [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex]. | [latex]{\Large\frac{1}{{x}^{3}}}[/latex] |
try it
[ohm_question]146245[/ohm_question]try it
[ohm_question]146299[/ohm_question]example
Simplify: 1. [latex]{\left(-3\right)}^{-2}[/latex] 2 [latex]{-3}^{-2}[/latex]Answer: Solution The negative in the exponent does not affect the sign of the base.
1. | |
The exponent applies to the base, [latex]-3[/latex] . | [latex]{\left(-3\right)}^{-2}[/latex] |
Take the reciprocal of the base and change the sign of the exponent. | [latex]{\Large\frac{1}{{\left(-3\right)}^{2}}}[/latex] |
Simplify. | [latex]{\Large\frac{1}{9}}[/latex] |
2. | |
The expression [latex]-{3}^{-2}[/latex] means: find the opposite of [latex]{3}^{-2}[/latex] The exponent applies only to the base, [latex]3[/latex]. | [latex]-{3}^{-2}[/latex] |
Rewrite as a product with [latex]−1[/latex]. | [latex]-1\cdot {3}^{-2}[/latex] |
Take the reciprocal of the base and change the sign of the exponent. | [latex]-1\cdot {\Large\frac{1}{{3}^{2}}}[/latex] |
Simplify. | [latex]-{\Large\frac{1}{9}}[/latex] |
try it
[ohm_question]146247[/ohm_question]example
Simplify: 1. [latex]4\cdot {2}^{-1}[/latex] 2. [latex]{\left(4\cdot 2\right)}^{-1}[/latex]Answer: Solution Remember to always follow the order of operations.
1. | |
Do exponents before multiplication. | [latex]4\cdot {2}^{-1}[/latex] |
Use [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex]. | [latex]4\cdot {\Large\frac{1}{{2}^{1}}}[/latex] |
Simplify. | [latex]2[/latex] |
2. | [latex]{\left(4\cdot 2\right)}^{-1}[/latex] |
Simplify inside the parentheses first. | [latex]{\left(8\right)}^{-1}[/latex] |
Use [latex]{a}^{-n}={\Large\frac{1}{{a}^{n}}}[/latex]. | [latex]{\Large\frac{1}{{8}^{1}}}[/latex] |
Simplify. | [latex]{\Large\frac{1}{8}}[/latex] |
try it
[ohm_question]146298[/ohm_question]example
Simplify: 1. [latex]5{y}^{-1}[/latex] 2. [latex]{\left(5y\right)}^{-1}[/latex] 3. [latex]{\left(-5y\right)}^{-1}[/latex]Answer: Solution
1. | |
Notice the exponent applies to just the base [latex]y[/latex] . | [latex]5{y}^{-1}[/latex] |
Take the reciprocal of [latex]y[/latex] and change the sign of the exponent. | [latex]5\cdot {\Large\frac{1}{{y}^{1}}}[/latex] |
Simplify. | [latex]{\Large\frac{5}{y}}[/latex] |
2. | |
Here the parentheses make the exponent apply to the base [latex]5y[/latex] . | [latex]{\left(5y\right)}^{-1}[/latex] |
Take the reciprocal of [latex]5y[/latex] and change the sign of the exponent. | [latex]{\Large\frac{1}{{\left(5y\right)}^{1}}}[/latex] |
Simplify. | [latex]{\Large\frac{1}{5y}}[/latex] |
3. | |
[latex]{\left(-5y\right)}^{-1}[/latex] | |
The base is [latex]-5y[/latex] . Take the reciprocal of [latex]-5y[/latex] and change the sign of the exponent. | [latex]{\Large\frac{1}{{\left(-5y\right)}^{1}}}[/latex] |
Simplify. | [latex]{\Large\frac{1}{-5y}}[/latex] |
Use [latex]{\Large\frac{a}{-b}}=-{\Large\frac{a}{b}}[/latex] | [latex]-{\Large\frac{1}{5y}}[/latex] |
try it
[ohm_question]146300[/ohm_question]Example
Write [latex]\frac{{\left({t}^{3}\right)}}{{\left({t}^{8}\right)}}[/latex] with positive exponents.
Answer:
Use the quotient rule to subtract the exponents of terms with like bases.
[latex]\begin{array}{r}\frac{{\left({t}^{3}\right)}}{{\left({t}^{8}\right)}}={t}^{3-8}\\={t}^{-5}\,\,\end{array}[/latex]
Write the expression with positive exponents by putting the term with the negative exponent in the denominator.
[latex]=\frac{1}{{t}^{5}}[/latex]
Answer
[latex-display]\frac{1}{{t}^{5}}[/latex-display]Example
Simplify [latex]{\left(\frac{1}{3}\right)}^{-2}[/latex].Answer: Apply the power property of exponents.
[latex]\frac{{1}^{-2}}{{3}^{-2}}[/latex]
Write each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.[latex]\frac{{3}^{2}}{{1}^{2}}{ = }\frac{{3}\cdot{3}}{{1}\cdot{1}}[/latex]
Simplify.[latex]\frac{{3}\cdot{3}}{{1}\cdot{1}}{ = }\frac{9}{1}{ = }{9}[/latex]
Answer
[latex-display]9[/latex-display]Example
Simplify.[latex]\frac{1}{4^{-2}}[/latex] Write your answer using positive exponents.Answer: Write each term with a positive exponent, the denominator will go to the numerator.
[latex]\frac{1}{4^{-2}}=1\cdot\frac{4^{2}}{1}=\frac{16}{1}=16[/latex]
Answer
[latex-display]16[/latex-display]Simplify Expressions with Zero Exponents
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like [latex]\Large\frac{{a}^{m}}{{a}^{m}}[/latex]. From earlier work with fractions, we know that[latex]\Large\frac{2}{2}\normalsize =\Large\frac{17}{17}\normalsize =\Large\frac{-43}{-43}\normalsize =1[/latex]
In words, a number divided by itself is [latex]1[/latex]. So [latex]\Large\frac{x}{x}\normalsize =1[/latex], for any [latex]x[/latex] ( [latex]x\ne 0[/latex] ), since any number divided by itself is [latex]1[/latex]. The Quotient Property of Exponents shows us how to simplify [latex]\Large\frac{{a}^{m}}{{a}^{n}}[/latex] by subtracting exponents. What if [latex]m=n[/latex] ? Now we will simplify [latex]\Large\frac{{a}^{m}}{{a}^{m}}[/latex] in two ways to lead us to the definition of the zero exponent. Consider first [latex]\Large\frac{8}{8}[/latex], which we know is [latex]1[/latex].[latex]\Large\frac{8}{8}\normalsize =1[/latex] | |
Write [latex]8[/latex] as [latex]{2}^{3}[/latex] . | [latex]\Large\frac{{2}^{3}}{{2}^{3}}\normalsize =1[/latex] |
Subtract exponents. | [latex]{2}^{3 - 3}=1[/latex] |
Simplify. | [latex]{2}^{0}=1[/latex] |
Exponents of 0 or 1
Any number or variable raised to a power of [latex]1[/latex] is the number itself.[latex]n^{1}=n[/latex]
Any non-zero number or variable raised to a power of [latex]0[/latex] is equal to [latex]1[/latex][latex]n^{0}=1[/latex]
The quantity [latex]0^{0}[/latex] is undefined.example
Simplify: 1. [latex]{12}^{0}[/latex] 2. [latex]{y}^{0}[/latex]Answer: Solution The definition says any non-zero number raised to the zero power is [latex]1[/latex].
1. | |
[latex]{12}^{0}[/latex] | |
Use the definition of the zero exponent. | [latex]1[/latex] |
2. | |
[latex]{y}^{0}[/latex] | |
Use the definition of the zero exponent. | [latex]1[/latex] |
try it
[ohm_question]146221[/ohm_question] [ohm_question]146890[/ohm_question][latex]{\left(2x\right)}^{0}[/latex] | |
Use the Product to a Power Rule. | [latex]{2}^{0}{x}^{0}[/latex] |
Use the Zero Exponent Property. | [latex]1\cdot 1[/latex] |
Simplify. | [latex]1[/latex] |
example
Simplify: [latex]{\left(7z\right)}^{0}[/latex].Answer: Solution
[latex]{\left(7z\right)}^{0}[/latex] | |
Use the definition of the zero exponent. | [latex]1[/latex] |
try it
[ohm_question]146222[/ohm_question]example
Simplify: 1. [latex]{\left(-3{x}^{2}y\right)}^{0}[/latex] 2. [latex]-3{x}^{2}{y}^{0}[/latex]Answer: Solution
1. | |
The product is raised to the zero power. | [latex]{\left(-3{x}^{2}y\right)}^{0}[/latex] |
Use the definition of the zero exponent. | [latex]1[/latex] |
2. | |
Notice that only the variable [latex]y[/latex] is being raised to the zero power. | [latex]{-3{x}^{2}y}^{0}[/latex] |
Use the definition of the zero exponent. | [latex]-3{x}^{2}\cdot 1[/latex] |
Simplify. | [latex]-3{x}^{2}[/latex] |
try it
[ohm_question]146223[/ohm_question] [ohm_question]146222[/ohm_question]Example
Evaluate [latex]2x^{0}[/latex] if [latex]x=9[/latex]Answer: Substitute 9 for the variable x.
[latex]2\cdot9^{0}[/latex]
Evaluate [latex]9^{0}[/latex]. Multiply.[latex]2\cdot1=2[/latex]
Answer
[latex-display]2x^{0}=2[/latex], if [latex]x=9[/latex-display]Example
Simplify [latex]\frac{{c}^{3}}{{c}^{3}}[/latex].Answer: Use the quotient and zero exponent rules to simplify the expression.
[latex]\begin{array}\text{ }\frac{c^{3}}{c^{3}} \,\,\,= \,\,\,c^{3-3} \\ \,\,\,\,\,\,\,\,\,\,\,=\,\,\,c^{0} \\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,1\end{array}[/latex]
Answer
[latex-display]1[/latex-display]Contribute!
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- Question ID: 146245, 146247, 146298, 146299, 146300. Authored by: Lumen Learning. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
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