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

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

Step 1

Simplify the numerator.

Apply the product rule to $x^{2} y^{3}$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Multiply the exponents in ${(y^{3})}^{2}$ .

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

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

Multiply $3$ by $2$ .

${(\frac{x^{4} y^{6}}{2 z})}^{3}$

Step 2

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

Apply the product rule to $\frac{x^{4} y^{6}}{2 z}$ .

$\frac{{(x^{4} y^{6})}^{3}}{{(2 z)}^{3}}$

Apply the product rule to $x^{4} y^{6}$ .

$\frac{{(x^{4})}^{3} {(y^{6})}^{3}}{{(2 z)}^{3}}$

Apply the product rule to $2 z$ .

$\frac{{(x^{4})}^{3} {(y^{6})}^{3}}{2^{3} z^{3}}$

Step 3

Simplify the numerator.

Multiply the exponents in ${(x^{4})}^{3}$ .

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

$\frac{x^{4 \cdot 3} {(y^{6})}^{3}}{2^{3} z^{3}}$

Multiply $4$ by $3$ .

$\frac{x^{12} {(y^{6})}^{3}}{2^{3} z^{3}}$

Multiply the exponents in ${(y^{6})}^{3}$ .

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

$\frac{x^{12} y^{6 \cdot 3}}{2^{3} z^{3}}$

Multiply $6$ by $3$ .

$\frac{x^{12} y^{18}}{2^{3} z^{3}}$

Step 4

Raise $2$ to the power of $3$ .

$\frac{x^{12} y^{18}}{8 z^{3}}$