Negative exponents and power rule for exponents
How to deal with negative exponents
This lesson will cover how to find the power of a negative exponent by using the power rule.
Case 1 of the power rule for negative exponents:
If you have two positive real numbers ???a??? and ???b??? then
???b^{-a} = \frac{1}{b^a}???
Think of it this way: in order to change the exponent in ???b^{-a}??? from ???-a??? to positive ???a??? you move the entire value from the numerator to the denominator to get
???\frac{1}{b^a}???
Case 2 of the power rule for negative exponents:
If you have two positive real numbers ???a??? and ???b??? then
???\frac{1}{b^{-a}}=b^a???
Think of it this way: in order to change the exponent in ???b^{-a}??? from ???-a??? to positive ???a??? you move the entire value from the denominator to the numerator to get ???1b^a??? which is the same as ???b^a???.
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 ???y^{-1}???. In order to change the exponent from ???-1??? to ???1??? you move the entire value from the numerator to the denominator to get
???\frac{1}{y^1}???
???\frac{1}{y}???
This means that ???y??? and ???y^{-1}??? are reciprocals.
How negative exponents and power rule for exponents are related to each other
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Rewriting expressions to eliminate the negative exponents
Example
Write the following without any negative exponents.
???2^{-1}???
In order to get rid of the negative exponent, we move the ???2^{-1}??? from the numerator to the denominator we get
???\frac{1}{2^1}???
Which is the same as
???\frac{1}{2}???
Let’s look at an example with a variable.
Example
Get rid of the negative exponents.
???x^{-5}???
In order to get rid of the negative exponent, we move the ???x^{-5}??? from the numerator to the denominator. We get
???\frac{1}{x^5}???
Let’s look at another example.
Example
Get rid of the negative exponents.
???\frac{1}{b^{-7}}???
In order to get rid of the negative exponent, we move the ???b^{-7}??? from the denominator to the numerator. We get ???1b^7??? which is the same as ???b^7???.
Let’s look at a final example with a number other than ???1??? in the numerator.
Example
Write the expression without negative exponents.
???\frac{3}{x^{-5}}???
In order to get rid of the negative exponent, we move the ???x^{-5}??? from the denominator to the numerator. We get
???3x^{5}???