Question

How do you rewrite ${{x}^{-5}}{{x}^{5}}$ using a positive exponent?

Answer 1

Hint: We should remember the identity ${{x}^{-n}}=\dfrac{1}{{{x}^{n}}}.$ We can always eliminate the common factor from the denominator and the numerator. Also, we should keep the fact that any number divided by the same number equals to $1.$

Complete step by step solution:
Consider the given expression ${{x}^{-5}}{{x}^{5}}.$
Let us recall the identity given by ${{x}^{-n}}=\dfrac{1}{{{x}^{n}}}.$
Here, we have a term with negative exponent in the given algebraic expression. That is, ${{x}^{-5}}.$
Using the above identity, we can write ${{x}^{-5}}=\dfrac{1}{{{x}^{5}}}.$
Now the expression will become ${{x}^{-5}}{{x}^{5}}=\dfrac{1}{{{x}^{5}}}{{x}^{5}}.$
That is, ${{x}^{-5}}{{x}^{5}}=\dfrac{{{x}^{5}}}{{{x}^{5}}}.$
We know that ${{x}^{5}}$ is a common factor in the numerator and the denominator. On the other hand, we can say that ${{x}^{5}}$ is divided by ${{x}^{5}}$ itself.
So, the quotient is $1.$
Now, we get the expression as ${{x}^{-5}}{{x}^{5}}=1={{x}^{0}},$ since ${{x}^{0}}=1.$
Or using the above identity ${{x}^{-n}}=\dfrac{1}{{{x}^{n}}}$ which implies ${{x}^{n}}=\dfrac{1}{{{x}^{-n}}},$ we can write the given algebraic expression as ${{x}^{-5}}{{x}^{5}}={{x}^{-5}}\dfrac{1}{{{x}^{-5}}}.$
And this can be written as ${{x}^{-5}}{{x}^{5}}=\dfrac{{{x}^{-5}}}{{{x}^{-5}}}.$
In the same way we have written above, we can write this as well.
So, let us eliminate ${{x}^{-5}}$ from both the numerator and the denominator.
So, we will get ${{x}^{-5}}{{x}^{5}}=1={{x}^{0}}.$
Hence the simplified form of the given algebraic expression ${{x}^{-5}}{{x}^{5}}$ using a positive exponent is $1$ where $1={{1}^{1}}.$

Note: There is an alternative method to simplify the given algebraic expression that is done using another familiar identity. The identity we just mentioned is ${{x}^{-n}}{{x}^{m}}={{x}^{-n+m}}.$ That is, ${{x}^{-n}}{{x}^{n}}={{x}^{-n+n}}={{x}^{0}}=1.$ When we are applying this identity in the given algebraic expression, we will get ${{x}^{-5}}{{x}^{5}}={{x}^{-5+5}}={{x}^{0}}=1.$