How To Change A Negative Exponent To A Positive

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 existent number, [latex]a\ne 0[/latex], and [latex]one thousand,due north[/latex] are whole numbers, then

[latex]\frac{{a}^{g}}{{a}^{due north}}={a}^{m-n},m>north\text{and}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},due north>k[/latex]

What if nosotros just subtract exponents, regardless of which is larger? Let'due south consider [latex]\frac{{ten}^{2}}{{x}^{5}}[/latex].
Nosotros subtract the exponent in the denominator from the exponent in the numerator.

[latex]\frac{{x}^{two}}{{x}^{five}}[/latex]
[latex]{x}^{two - 5}[/latex]
[latex]{x}^{-three}[/latex]
We tin can too simplify [latex]\frac{{x}^{2}}{{x}^{5}}[/latex] by dividing out common factors: [latex]\frac{{x}^{ii}}{{x}^{five}}[/latex].

$A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.$
This implies that [latex]{ten}^{-three}=\frac{ane}{{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], and so [latex]{a}^{-n}=\frac{one}{{a}^{n}}[/latex].

The negative exponent tells us to re-write the expression past taking the reciprocal of the base and so changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. Nosotros will utilize the definition of a negative exponent and other properties of exponents to write an expression with but positive exponents.

example

Simplify:

1. [latex]{four}^{-two}[/latex]
2. [latex]{10}^{-3}[/latex]

Solution

1.
	[latex]{4}^{-2}[/latex]
Use the definition of a negative exponent, [latex]{a}^{-north}=\frac{ane}{{a}^{n}}[/latex].	[latex]\frac{i}{{4}^{ii}}[/latex]
Simplify.	[latex]\frac{1}{16}[/latex]

2.
	[latex]{10}^{-3}[/latex]
Employ the definition of a negative exponent, [latex]{a}^{-n}=\frac{i}{{a}^{due north}}[/latex].	[latex]\frac{1}{{x}^{iii}}[/latex]
Simplify.	[latex]\frac{1}{1000}[/latex]

try it

When simplifying whatever expression with exponents, nosotros must be conscientious to correctly identify the base that is raised to each exponent.

example

Simplify:

ane. [latex]{\left(-3\correct)}^{-2}[/latex]
ii [latex]{-3}^{-two}[/latex]

endeavor it

We must be careful to follow the social club of operations. In the side by side example, parts 1 and ii wait similar, but we go dissimilar results.

example

Simplify:

1. [latex]4\cdot {two}^{-1}[/latex]
2. [latex]{\left(four\cdot 2\right)}^{-1}[/latex]

try it

When a variable is raised to a negative exponent, we apply the definition the aforementioned way nosotros did with numbers.

instance

Simplify: [latex]{ten}^{-6}[/latex].

try information technology

When there is a production and an exponent nosotros 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. Nosotros'll meet how this works in the next example.

case

Simplify:

1. [latex]v{y}^{-1}[/latex]
ii. [latex]{\left(5y\right)}^{-ane}[/latex]
iii. [latex]{\left(-5y\right)}^{-1}[/latex]

try it

VIDEO Asking

Now that nosotros have divers negative exponents, the Quotient Property of Exponents needs only one form, [latex]\frac{{a}^{chiliad}}{{a}^{n}}={a}^{m-north}[/latex], where [latex]a\ne 0[/latex] and g and north are integers.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will exist negative. If the event gives us a negative exponent, we volition rewrite information technology past using the definition of negative exponents, [latex]{a}^{-northward}=\frac{ane}{{a}^{n}}[/latex].