Question

How do you simply \[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}}\] ?

Answer 1

Hint: In this question, we are given a fraction in which both the numerator and the denominator are exponential functions and we have to simplify this fraction. For that, we will use the law of exponents which states that the ratio of two exponential functions having the same base is equal to that base raised to the power the difference of the powers of the numerator and the denominator, that is, $\dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}}$ . This way we will get the given fraction in simplified form.

Complete step-by-step solution:
We have to simplify \[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}}\]
We know that –
$
  \dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}} \\
   \Rightarrow \dfrac{{{m^4}}}{{{m^2}}} = {m^{4 - 2}} = {m^2} \\
   \Rightarrow \dfrac{{{n^3}}}{{{n^5}}} = {n^{3 - 5}} = {n^{ - 2}} \\
 $
So, we get –
\[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}} = \dfrac{{36}}{{24}}{m^2}{n^{ - 2}}\]
Now we see that 36 and 24 are multiples of 12, so 12 is canceled out and we get –
\[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}} = \dfrac{3}{2}{m^2}{n^{ - 2}}\]
We know that ${a^{ - x}} = \dfrac{1}{{{a^x}}}$ , so we can also write \[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}}\] as $\dfrac{{3{m^2}}}{{2{n^2}}}$
Hence, the simplified form of \[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}}\] is $\dfrac{3}{2}{m^2}{n^{ - 2}}$ or $\dfrac{{3{m^2}}}{{2{n^2}}}$ .

Note: The expression given in the question is a fraction and we know that we write the numerator and the denominator as a product of its prime factors for simplifying it. After doing the prime factorization of the numerator and the denominator, we cancel out the common factors until there are no common factors present between the numerator and the denominator.
${m^4}$ can be written as $m \times m \times m \times m$ , ${m^2}$ can be written as \[m \times m\] , ${n^3}$ can be written as $n \times n \times n$ , ${n^5}$ can be written as $n \times n \times n \times n \times n$ , $36 = 2 \times 2 \times 3 \times 3 = 12 \times 3$ and $24 = 2 \times 2 \times 2 \times 3 = 12 \times 2$ .Thus we see that the numerator and the denominator have ${m^2},\,{n^3}\,and\,12$ as common, so it is canceled out and we get –
\[\dfrac{{36{m^4}{n^3}}}{{24{m^2}{n^5}}}\] is equal to $\dfrac{{3{m^2}}}{{2{n^2}}}\,or\,\dfrac{3}{2}{m^2}{n^{ - 2}}$