Question

Find the value of $n,$ if $\dfrac{{{6}^{n}}}{{{6}^{-2}}}={{6}^{3}}.$

Answer 1

Hint: We are going to use the identity that says $\dfrac{1}{{{x}^{-n}}}={{x}^{n}}.$ Also, we will use another identity that deals with the exponents of the same value given by ${{x}^{n}}{{x}^{m}}={{x}^{n+m}}.$ Also, we know the fact that ${{x}^{m}}={{x}^{n}}$ implies $n=m.$

Complete step-by-step answer:
We are asked to find the value of $n$ when it is given that $\dfrac{{{6}^{n}}}{{{6}^{-2}}}={{6}^{3}}.$
Let us consider the given information.
We need to simplify the left-hand side of the given equation so that the further procedures become easy. So, let us focus on the left-hand side of the equation, $\dfrac{{{6}^{n}}}{{{6}^{-2}}}.$
Now, we know that $\dfrac{1}{{{x}^{-n}}}={{x}^{n}}.$
We are going to use this identity to make the terms on the left-hand side the terms having only positive exponents.
We can see that the denominator on the left-hand side is having a negative exponent.
When we use the above identity, we will get $\dfrac{1}{{{6}^{-2}}}={{6}^{2}}.$
So, the given equation will become ${{6}^{n}}{{6}^{2}}={{6}^{3}}.$
Since the terms on the left-hand side of the equations containing the same number $6$ with different exponents, we can use the identity ${{x}^{n}}{{x}^{m}}={{x}^{n+m}}.$
So, now we can write the left-hand side of the equation as ${{6}^{n}}{{6}^{2}}={{6}^{n+2}}.$
Now, we will write the equation with the obtained value.
So, the equation will become ${{6}^{n+2}}={{6}^{3}}.$
We know that if ${{n}^{th}}$ power of a number is equal to ${{m}^{th}}$ power of the number, then $m=n.$
We can say that if ${{x}^{n}}={{x}^{m}},$ then $n=m.$
Therefore, we can conclude that $n+2=3.$
Let us transpose $2$ from the left-hand side of the equation to the right-hand side of the equation to get the required value.
We will get $n=3-2=1.$
Hence $n=1.$

Note: We can find the value of $n$ without making the whole equation having only positive exponents. We can transpose the term in the denominator of the fraction on the left-hand side to the right-hand side to get ${{6}^{n}}={{6}^{3}}{{6}^{-2}}.$ Now, we will get \[{{6}^{n}}={{6}^{3-2}}={{6}^{1}}.\] Therefore, $n=1.$