Question

How do you simplify \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\]\[\]

Answer 1

Hint: We are given a fraction as \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\], we are asked to simplify, we will first learn about the type of fraction once we find the type of fraction then we will learn about the techniques which will help us to solve and simplify that particular form of fraction problems. We have \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\]in which \[{{3}^{7}}\] is an exponential form of term so we get it in exponential form of fraction we will expand the term and simplify to get our answer.

Complete step by step answer:
We are given a fraction as \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\] we are asked to simplify this. To do simplification we will learn about how we simplify any rational expression. To do so we will first learn what type of rational expression we are dealing with then we work according to the type.
Now we have \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\], we can see that numerator is \[{{3}^{7}}\] and denominator is \[{{3}^{2}}\] both are in form of exponential \[{{a}^{b}}\]. So here we are dealing with an exponential form of rational. To simplify we will expand the terms in numerator and denominator and choose for the term when can be cancelled.
For example, \[\dfrac{{{2}^{3}}}{{{4}^{2}}}\]so we expand numerator and denominator so we get \[\dfrac{{{2}^{3}}}{{{4}^{2}}}=\dfrac{2\times 2\times 2}{4\times 4}\], Now cancelling terms, we get after simplification.
\[\dfrac{{{2}^{3}}}{{{4}^{2}}}=\dfrac{1}{2}\]
Now after this we will simplify by solving numerator and denominator in denominator, we have \[2\]. So our solution is
\[\dfrac{{{2}^{3}}}{{{4}^{2}}}=\dfrac{1}{2}\]
Now in our equation \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\], we have numerator as \[{{3}^{7}}\] and denominator is \[{{3}^{2}}\] so simplifying we get
\[\dfrac{{{3}^{7}}}{{{3}^{2}}}=\dfrac{3\times 3\times 3\times 3\times 3\times 3\times 3}{3\times 3}\]
Cancelling like terms, we get
\[\dfrac{{{3}^{7}}}{{{3}^{2}}}=\dfrac{3\times 3\times 3\times 3\times 3}{1}\]
Now we solve numerator and denominator we get as \[3\times 3\times 3\times 3\times 3={{3}^{5}}=243\]and denominator as $1$ so, we get
\[\dfrac{{{3}^{7}}}{{{3}^{2}}}=243\]
Hence we get simplified fraction of exponential type of \[\dfrac{{{3}^{7}}}{{{3}^{2}}}=243\] another way to solve 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 same base then the power of those are substantial that in \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\]
So we have our fraction as \[\dfrac{{{3}^{7}}}{{{3}^{2}}}\] so using above formulas, we get \[\dfrac{{{3}^{7}}}{{{3}^{2}}}={{3}^{7-2}}\], which is simplified as \[\dfrac{{{3}^{7}}}{{{3}^{2}}}={{3}^{5}}=243\]

So, simplification value of \[\dfrac{{{3}^{7}}}{{{3}^{2}}}=243\]

Note: While solving exponential problem we need to be careful that we have \[{{3}^{2}}\] it means that \[{{3}^{2}}=3\times 3\] and it is not \[{{3}^{2}}\ne 3\times 2\]. Many errors like \[{{3}^{7}}\ne 3\times 7\]happen, Remember that term on the head of $3$ is its power we have to multiply $3,7$ times we do not have to multiply $3$with $7$ if we do that we will never get correct solution.