What to do with negative exponents

 
 
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How do you deal with negative exponents?

This lesson will cover how to find the power of a negative exponent.

 

Part 1: A reminder

Remember that any number can be written as itself divided by ???1???. For example, ???3??? is the same as ???3/1???. Also remember that the top part of a fraction is called the numerator and the bottom part of a fraction is called the denominator.

 

Part 2: The rule for negative exponents

If you have two positive real numbers ???a??? and ???b???, then

???b^{-a} = \frac{1}{b^a}???

You can think of it like this: first we need to realize that ???b^{-a}??? is the same as

???\frac{b^{-a}}{1}???

We’ll change the exponent in ???b^{-a}??? from ???-a??? to ???a??? by moving the entire value from the numerator to the denominator to get the

???\frac{1}{b^a}???

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Let's take a brief timeout to talk about reciprocals

By the way, ???a^b??? and ???1/a^b??? are called “reciprocals”. Sometimes you’ll hear or read about negative exponents and their relationship to reciprocals and that’s because of this relationship.

Think about ???4^{-1}???. First realize that ???4^{-1}??? is the same as

???\frac{4^{-1}}{1}???

We’ll change the exponent in ???4^{-1}??? from ???-1??? to ???1??? by moving the entire value from the numerator to the denominator to get

???\frac{1}{4^{1}}???

???\frac{1}{4}???

 
 
 

This is a video with lots of examples of how to change negative exponents to positive exponents and vice versa


 
 
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An example of changing a negative exponent into a positive exponent

Example

Simplify the expression.

???4^{-2}???

 

Remember that ???4^{-2}??? is the same as

???\frac{4^{-2}}{1}???

We’ll change the exponent from ???-2??? to ???2??? by moving the entire value from the numerator to the denominator to get,

???\frac{1}{4^2}???

Now we’ll perform the calculation in the denominator.

???\frac{1}{4^2} = \frac{1}{4 \cdot 4}??? ???= \frac{1}{16}???

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Moving a term from the numerator to the denominator, or vice versa, changes the sign of the exponent

 

An example with a negative sign in front of the base

Example

Simplify the expression.

???-5^{-3}???

 

Remember, we can rewrite ???-5^{-3}??? as

???\frac{-5^{-3}}{1}???

because they are the same value.

We’ll change the exponent from ???-3??? to ???3??? by moving the entire value from the numerator to the denominator.

???\frac{1}{-5^3}???

We have to apply the exponent before we apply the negative sign so the expression becomes

???\frac{1}{-125}???

???-\frac{1}{125}???

 

A negative exponent when the base is a variable

Example

Write the expression with only positive exponents.

???x^{-3}???

 

First, we need to realize that the expression ???x^{-3}??? is the same as

???\frac{x^{-3}}{1}???

We’ll change the exponent from ???-3??? to ???3??? by moving the entire value from the numerator to the denominator.

???\frac{1}{x^3}???

 
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