Writing Negative Exponents as Positive Exponents
Learning Outcomes
- Simplify expressions with negative exponents
The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.
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]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\text{and}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},n>m[/latex-display]
What if we just subtract exponents, regardless of which is larger? Let’s consider [latex]\frac{{x}^{2}}{{x}^{5}}[/latex].
We subtract the exponent in the denominator from the exponent in the numerator.
[latex-display]\frac{{x}^{2}}{{x}^{5}}[/latex-display]
[latex-display]{x}^{2 - 5}[/latex-display]
[latex-display]{x}^{-3}[/latex-display]
We can also simplify [latex]\frac{{x}^{2}}{{x}^{5}}[/latex] by dividing out common factors: [latex]\frac{{x}^{2}}{{x}^{5}}[/latex].
This implies that [latex]{x}^{-3}=\frac{1}{{x}^{3}}[/latex] and it leads us to the definition of a negative exponent.
Negative Exponent
If [latex]n[/latex] is a positive integer and [latex]a\ne 0[/latex], then [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex].
The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
example
Simplify:
1. [latex]{4}^{-2}[/latex]
2. [latex]{10}^{-3}[/latex]
Solution
| 1. |
|
|
[latex]{4}^{-2}[/latex] |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]\frac{1}{{4}^{2}}[/latex] |
| Simplify. |
[latex]\frac{1}{16}[/latex] |
| 2. |
|
|
[latex]{10}^{-3}[/latex] |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]\frac{1}{{10}^{3}}[/latex] |
| Simplify. |
[latex]\frac{1}{1000}[/latex] |
try it
[ohm_question]146245[/ohm_question]
When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.
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]\frac{1}{{\left(-3\right)}^{2}}[/latex] |
| Simplify. |
[latex]\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 \frac{1}{{3}^{2}}[/latex] |
| Simplify. |
[latex]-\frac{1}{9}[/latex] |
try it
[ohm_question]146247[/ohm_question]
We must be careful to follow the order of operations. In the next example, parts 1 and 2 look similar, but we get different results.
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}=\frac{1}{{a}^{n}}[/latex]. |
[latex]4\cdot \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}=\frac{1}{{a}^{n}}[/latex]. |
[latex]\frac{1}{{8}^{1}}[/latex] |
| Simplify. |
[latex]\frac{1}{8}[/latex] |
try it
[ohm_question]146298[/ohm_question]
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.
example
Simplify: [latex]{x}^{-6}[/latex].
Answer:
Solution
|
[latex]{x}^{-6}[/latex] |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. |
[latex]\frac{1}{{x}^{6}}[/latex] |
try it
[ohm_question]146299[/ohm_question]
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.
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 \frac{1}{{y}^{1}}[/latex] |
| Simplify. |
[latex]\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]\frac{1}{{\left(5y\right)}^{1}}[/latex] |
| Simplify. |
[latex]\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]\frac{1}{{\left(-5y\right)}^{1}}[/latex] |
| Simplify. |
[latex]\frac{1}{-5y}[/latex] |
| Use [latex]\frac{a}{-b}=-\frac{a}{b}[/latex]. |
[latex]-\frac{1}{5y}[/latex] |
try it
[ohm_question]146300[/ohm_question]
VIDEO REQUEST
Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], where [latex]a\ne 0[/latex] and m and n are integers.
When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex].Licenses & Attributions
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