# How do you simplify (5x^2y^3)^2*(2x^3y^4)^3 and write it using only positive exponents?

Jan 2, 2018

$200 {x}^{13} {y}^{18}$

#### Explanation:

$\text{by appling the "color(blue)"laws of exponents}$

•color(white)(x)(a^m)^(n)=a^((mxxn))larr(color(red)(1))

•color(white)(x)a^mxxa^n=a^((m+n))

$\left(\textcolor{red}{1}\right) \text{ is extended to include all factors inside the parenthesis}$

$\Rightarrow {\left(5 {x}^{2} {y}^{3}\right)}^{2} = {5}^{\left(1 \times 2\right)} \times {x}^{\left(2 \times 2\right)} \times {y}^{\left(3 \times 2\right)}$

$\textcolor{w h i t e}{\times \times \times \times} = {5}^{2} \times {x}^{4} \times {y}^{6} = 25 {x}^{4} {y}^{6}$

$\Rightarrow {\left(2 {x}^{3} {y}^{4}\right)}^{3} = {2}^{\left(1 \times 3\right)} \times {x}^{\left(3 \times 3\right)} \times {y}^{\left(4 \times 3\right)}$

$\textcolor{w h i t e}{\times \times \times \times} = {2}^{3} \times {x}^{9} \times {y}^{12} = 8 {x}^{9} {y}^{12}$

$\Rightarrow {\left(5 {x}^{2} {y}^{3}\right)}^{2} \times {\left(2 {x}^{3} {y}^{4}\right)}^{3}$

$= 25 {x}^{4} {y}^{6} \times 8 {x}^{9} {y}^{12}$

$= \left(25 \times 8\right) \times \left({x}^{4} \times {x}^{9}\right) \times \left({y}^{6} \times {y}^{12}\right)$

$= 200 \times {x}^{\left(4 + 9\right)} \times {y}^{\left(6 + 12\right)}$

$= 200 {x}^{13} {y}^{18}$

Jan 2, 2018

See a solution process below:

#### Explanation:

First, use these rules for exponents to eliminate the outer exponents for each term:

or $a = {a}^{\textcolor{red}{1}}$ and ${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left(5 {x}^{2} {y}^{3}\right)}^{2} \cdot {\left(2 {x}^{3} {y}^{4}\right)}^{3} \implies$

${\left({5}^{\textcolor{red}{1}} {x}^{\textcolor{red}{2}} {y}^{\textcolor{red}{3}}\right)}^{\textcolor{b l u e}{2}} \cdot {\left({2}^{\textcolor{red}{1}} {x}^{\textcolor{red}{3}} {y}^{\textcolor{red}{4}}\right)}^{\textcolor{b l u e}{3}} \implies$

$\left({5}^{\textcolor{red}{1} \times \textcolor{b l u e}{2}} {x}^{\textcolor{red}{2} \times \textcolor{b l u e}{2}} {y}^{\textcolor{red}{3} \times \textcolor{b l u e}{2}}\right) \cdot \left({2}^{\textcolor{red}{1} \times \textcolor{b l u e}{3}} {x}^{\textcolor{red}{3} \times \textcolor{b l u e}{3}} {y}^{\textcolor{red}{4} \times \textcolor{b l u e}{3}}\right) \implies$

$\left({5}^{2} {x}^{4} {y}^{6}\right) \cdot \left({2}^{3} {x}^{9} {y}^{12}\right) \implies$

$\left(25 {x}^{4} {y}^{6}\right) \cdot \left(8 {x}^{9} {y}^{12}\right)$

Next, rewrite the expression as:

$\left(25 \cdot 8\right) \left({x}^{4} \cdot {x}^{9}\right) \left({y}^{6} \cdot {y}^{12}\right) \implies$

$200 \left({x}^{4} \cdot {x}^{9}\right) \left({y}^{6} \cdot {y}^{12}\right)$

Now, use this rule of exponents to complete the simplification:

${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$

$200 \left({x}^{\textcolor{red}{4}} \cdot {x}^{\textcolor{b l u e}{9}}\right) \left({y}^{\textcolor{red}{6}} \cdot {y}^{\textcolor{b l u e}{12}}\right) \implies$

$200 {x}^{\textcolor{red}{4} + \textcolor{b l u e}{9}} {y}^{\textcolor{red}{6} + \textcolor{b l u e}{12}} \implies$

$200 {x}^{13} {y}^{18}$