# How do you multiply and simplify \frac { ( x ^ { 3} y ^ { 10} ) ( x ^ { 4} y ) } { ( x ^ { 8} y ^ { 2} ) ^ { 3} }?

Jun 21, 2018

$\frac{\left({x}^{3} {y}^{10}\right) \left({x}^{4} y\right)}{{\left({x}^{8} {y}^{2}\right)}^{3}}$

Remove the brackets in the numerator

$\frac{{x}^{7} {y}^{11}}{{\left({x}^{8} {y}^{2}\right)}^{3}}$

Remove the brackets in the denominator

$\frac{{x}^{7} {y}^{11}}{{x}^{24} {y}^{6}}$

cancel by ${x}^{7}$

${y}^{11} / \left[{x}^{17} {y}^{6}\right]$

cancel by ${y}^{6}$

${y}^{5} / {x}^{17}$

Jun 21, 2018

${y}^{5} / {x}^{17} \textcolor{w h i t e}{\ldots}$ Lot of explanation given

#### Explanation:

Given: $\frac{\left({x}^{3} {y}^{10}\right) \left({x}^{4} y\right)}{{\left({x}^{8} {y}^{2}\right)}^{3}}$

$\textcolor{b l u e}{\text{The numerator}}$

Consider the example: $\textcolor{w h i t e}{\text{d")2^2xx2^3color(white)("d")=color(white)("d}} 4 \times 8 = 32$
But this is the same as : $\textcolor{w h i t e}{\text{.d")2^(2+3)color(white)(".d")=color(white)("ddd")2^5color(white)(".}} = 32$

Applying this to the numerator $\left({x}^{3} {y}^{10}\right) \left({x}^{4} y\right)$

Write as: $\left[{x}^{3} \times {x}^{4}\right] \left[{y}^{10} \times {y}^{1}\right] = {x}^{3 + 4} \times {y}^{10 + 1} = \textcolor{red}{{x}^{7} {y}^{11}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{The denominator}}$

Consider the example:
${\left({2}^{4}\right)}^{3} \textcolor{w h i t e}{\text{d")=color(white)("d")2^4xx2^4xx2^4color(white)("d")=2^(4+4+4)=color(white)("d}} {2}^{4 \times 3} = {2}^{12}$

Applying this to the denominator ${\left({x}^{8} {y}^{2}\right)}^{3}$

Write as: ${x}^{8 \times 3} \times {y}^{2 \times 3} = \textcolor{g r e e n}{{x}^{24} {y}^{6}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

$\frac{\left({x}^{3} {y}^{10}\right) \left({x}^{4} y\right)}{{\left({x}^{8} {y}^{2}\right)}^{3}} = \frac{\textcolor{red}{{x}^{7} {y}^{11}}}{\textcolor{g r e e n}{{x}^{24} {y}^{6}}} \textcolor{w h i t e}{\text{d") = color(white)("ddd")x^7/x^24color(white)("dd")xxcolor(white)("ddd}} {y}^{11} / {y}^{6}$

$\textcolor{w h i t e}{\text{dddddddddddddddddd}} = \left[{x}^{7} / {x}^{7} \times \frac{1}{x} ^ 17\right] \times \left[{y}^{6} / {y}^{6} \times {y}^{5} / 1\right]$

$\textcolor{w h i t e}{\text{dddddddddddddddddd")=color(white)("ddd")[1/x^17]color(white)("dd")xxcolor(white)("dd}} \left[{y}^{5} / 1\right]$

$\textcolor{w h i t e}{\text{dddddddddddddddddd")=color(white)("dddddddddd}} {y}^{5} / {x}^{17}$