Simplify Using Only Positive Exponents

Learning Outcomes

Simplify expressions with negative exponents

The Quotient Property of Exponents has two forms depending on whether the exponent in the numerator or denominator was larger.

Quotient Belongings 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]{\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n\text{ and }{\Big\frac{{a}^{m}}{{a}^{n}}}={\Big\frac{ane}{{a}^{n-chiliad}}},n>thou[/latex]

What if we just subtract exponents, regardless of which is larger? Permit's consider [latex]{\Large\frac{{x}^{2}}{{x}^{5}}}[/latex]
We subtract the exponent in the denominator from the exponent in the numerator.

[latex]{\Big\frac{{x}^{2}}{{x}^{five}}}[/latex] [latex]=[/latex] [latex]{ten}^{2 - 5}[/latex] [latex]=[/latex] [latex]{ten}^{-iii}[/latex]

We tin also simplify [latex]{\Big\frac{{ten}^{two}}{{x}^{5}}}[/latex] past dividing out mutual factors:

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

The negative exponent tells united states to re-write the expression past taking the reciprocal of the base and so irresolute the sign of the exponent. Any expression that has negative exponents is not considered to exist in simplest form. Nosotros volition use the definition of a negative exponent and other properties of exponents to write an expression with merely positive exponents.

example

Simplify:

1. [latex]{4}^{-2}[/latex]
2. [latex]{10}^{-three}[/latex]

Solution

i.
	[latex]{4}^{-two}[/latex]
Use the definition of a negative exponent, [latex]{a}^{-due north}={\Large\frac{1}{{a}^{due north}}}[/latex].	[latex]{\Big\frac{i}{{iv}^{2}}}[/latex]
Simplify.	[latex]{\Big\frac{1}{16}}[/latex]

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

try it

When simplifying any expression with exponents, we must exist conscientious 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]

try it

We must be careful to follow the order of operations. In the next example, parts i and two wait like, but we become different results.

example

Simplify:

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

try it

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

case

Simplify: [latex]{x}^{-6}[/latex]

try it

When there is a product and an exponent we have to be careful to use the exponent to the right quantity. Co-ordinate to the order of operations, expressions in parentheses are simplified before exponents are practical. We'll see how this works in the next example.

example

Simplify:

1. [latex]5{y}^{-1}[/latex]
2. [latex]{\left(5y\right)}^{-one}[/latex]
iii. [latex]{\left(-5y\correct)}^{-1}[/latex]

endeavour it

Now that we take defined negative exponents, the Quotient Property of Exponents needs only one class, [latex]{\Large\frac{{a}^{yard}}{{a}^{n}}}={a}^{thousand-north}[/latex], where [latex]a\ne 0[/latex] and thou 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 consequence gives us a negative exponent, we will rewrite it by using the definition of negative exponents, [latex]{a}^{-n}={\Large\frac{i}{{a}^{north}}}[/latex].