# How do you simplify (\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }?

May 3, 2018

Here's how it's done

#### Explanation:

Cancel common factor $x$
$= {x}^{6} {y}^{3}$
$\setminus = {\left({x}^{6} {y}^{3} \setminus\right)}^{\setminus \frac{1}{3}}$

Apply the exponent rule
$= \setminus {\left({x}^{6} \setminus\right)}^{\setminus \frac{1}{3}} \setminus {\left({y}^{3} \setminus\right)}^{\setminus \frac{1}{3}}$

Final solution
$= {x}^{2} y$

May 4, 2018

$\text{ }$
color(blue)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }=x^2y

#### Explanation:

$\text{ }$
Given the expression: color(red)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }

$\text{Exponents formula required: }$

color(blue)( a^m/a^n = a^(m-n)

color(blue)( a^m*a^n = a^(m+n)

color(blue)(( a^m)^(1/n) = a^(m/n)

color(green)("Step 1:"

Rewrite the given expression: color(brown)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} } as

[[(x^7*y^3)^(1/3)]]/[[x^(1/3)]

Rewrite as:

$\frac{{\left({x}^{7}\right)}^{\frac{1}{3}} \cdot {\left({y}^{3}\right)}^{\frac{1}{3}}}{x} ^ \left(\frac{1}{3}\right)$

color(green)("Step 2:"

Use: color(blue)(( a^m)^(1/n) = a^(m/n)

Simplify:

$\frac{{x}^{\frac{7}{3}} \cdot {y}^{\frac{3}{3}}}{x} ^ \left(\frac{1}{3}\right)$

color(green)("Step 3:"

$\frac{{x}^{\frac{7}{3}} \cdot {y}^{1}}{x} ^ \left(\frac{1}{3}\right)$

color(green)("Step 4:"

Use: color(blue)( a^m/a^n = a^(m-n)

$\left[{x}^{\frac{7}{3}} - {x}^{\frac{1}{3}}\right] \cdot y$

${x}^{\frac{6}{3}} \cdot y$

$\Rightarrow {x}^{2} \cdot y$

Hence,

color(blue)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }=x^2y

Hope it helps.