Question

How do you simplify \[\dfrac{{{8}^{2}}}{{{8}^{4}}}?\]

Answer 1

Hint: We are given a fraction as \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] and we are asked to simplify. To simplify we will first learn about the type of the fraction. Once we find the type of the fraction then we will learn about the techniques which will help us to solve and simplify that particular form of fraction problems. We have \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] in which \[{{8}^{2}}\] is an exponential form of the term. So, we get it in exponential form of the fraction. We will expand the term and simplify to get our answer.

Complete step by step answer:
We are given a fraction as \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] and we are asked to simplify this. To do the simplification, we will learn about the way we simplify any rational expression. To do so we will first learn what type of rational expression we are dealing with and then we will work according to the type. Now we have \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] and we see that the numerator is \[{{8}^{2}}\] and the denominator is \[{{8}^{4}}\] and both are in the form of exponential \[{{a}^{b}}.\] So, here we are dealing with the exponential form of the rational. To simplify we will expand the terms in the numerator and denominator and check for the term which can be cancelled. For example, \[\dfrac{{{2}^{3}}}{{{4}^{2}}}\] here we expand the numerator and denominator. So, we get,
\[\dfrac{{{2}^{3}}}{{{4}^{2}}}=\dfrac{2\times 2\times 2}{4\times 4}\]
Now, cancelling the terms and after simplification we get,
\[\Rightarrow \dfrac{{{2}^{3}}}{{{4}^{2}}}=\dfrac{1}{2\times 4}\]
Now, after this we will simplify by solving the numerator and denominator. In the denominator, we have \[2\times 4\] and we get 8. So, our solution is
\[\Rightarrow \dfrac{{{2}^{3}}}{{{4}^{2}}}=\dfrac{1}{8}\]
Now in our equation, \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] we have numerator as \[{{8}^{2}}\] and denominator as \[{{8}^{4}}.\] So, expanding we get,
\[\dfrac{{{8}^{2}}}{{{8}^{4}}}=\dfrac{8\times 8}{8\times 8\times 8\times 8}\]
Cancelling like terms, we get,
\[\Rightarrow \dfrac{{{8}^{2}}}{{{8}^{4}}}=\dfrac{1}{8\times 8}\]
Now, after we solve the numerator and denominator. We get the numerator as 1 and denominator as \[8\times 8={{8}^{2}}=64.\] So, we get, \[\dfrac{{{8}^{2}}}{{{8}^{4}}}=\dfrac{1}{64}.\]
Hence, we get the simplified value of \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] is \[\dfrac{1}{64}.\]

Note: Another way to solve the fraction of exponential type of the form \[\dfrac{{{a}^{b}}}{{{a}^{c}}}\] is to use algebraic operation. We know when we divide two exponential with the same base then the power of those are subtracted that is \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}.\] So as we have our fraction as \[\dfrac{{{8}^{2}}}{{{8}^{4}}},\] so using the above identity we get \[\dfrac{{{8}^{2}}}{{{8}^{4}}}={{8}^{2-4}}\] which is simplified \[\dfrac{{{8}^{2}}}{{{8}^{4}}}={{8}^{-2}}=\dfrac{1}{{{8}^{2}}}\left[ \text{As }{{a}^{-b}}=\dfrac{1}{{{a}^{b}}} \right].\] So, the simplified form of \[\dfrac{{{8}^{2}}}{{{8}^{4}}}\] is \[\dfrac{1}{64}.\]