Simplify (x/y)^3((y^2x)/z)^4

${(\frac{x}{y})}^{3} {(\frac{y^{2} x}{z})}^{4}$

Step 1

Apply the product rule to $\frac{x}{y}$ .

$\frac{x^{3}}{y^{3}} {(\frac{y^{2} x}{z})}^{4}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{y^{2} x}{z}$ .

$\frac{x^{3}}{y^{3}} \cdot \frac{{(y^{2} x)}^{4}}{z^{4}}$

Apply the product rule to $y^{2} x$ .

$\frac{x^{3}}{y^{3}} \cdot \frac{{(y^{2})}^{4} x^{4}}{z^{4}}$

Step 3

Combine.

$\frac{x^{3} ({(y^{2})}^{4} x^{4})}{y^{3} z^{4}}$

Step 4

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$\frac{x^{4} x^{3} {(y^{2})}^{4}}{y^{3} z^{4}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{4 + 3} {(y^{2})}^{4}}{y^{3} z^{4}}$

Add $4$ and $3$ .

$\frac{x^{7} {(y^{2})}^{4}}{y^{3} z^{4}}$

Step 5

Multiply the exponents in ${(y^{2})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{7} y^{2 \cdot 4}}{y^{3} z^{4}}$

Multiply $2$ by $4$ .

$\frac{x^{7} y^{8}}{y^{3} z^{4}}$

Step 6

Cancel the common factor of $y^{8}$ and $y^{3}$ .

Factor $y^{3}$ out of $x^{7} y^{8}$ .

$\frac{y^{3} (x^{7} y^{5})}{y^{3} z^{4}}$

Cancel the common factors.

Factor $y^{3}$ out of $y^{3} z^{4}$ .

$\frac{y^{3} (x^{7} y^{5})}{y^{3} (z^{4})}$

Cancel the common factor.

$\frac{y^{3} (x^{7} y^{5})}{y^{3} z^{4}}$

Rewrite the expression.

$\frac{x^{7} y^{5}}{z^{4}}$