Simplify ((3x^2y^6)/(4y^4z))^-3

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

Step 1

Cancel the common factor of $y^{6}$ and $y^{4}$ .

Factor $y^{4}$ out of $3 x^{2} y^{6}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 3

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

Apply the product rule to $\frac{4 z}{3 x^{2} y^{2}}$ .

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

Apply the product rule to $4 z$ .

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

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

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

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

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

Step 4

Raise $4$ to the power of $3$ .

$\frac{64 z^{3}}{3^{3} {(x^{2})}^{3} {(y^{2})}^{3}}$

Step 5

Simplify the denominator.

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

$\frac{64 z^{3}}{27 {(x^{2})}^{3} {(y^{2})}^{3}}$

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

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

$\frac{64 z^{3}}{27 x^{2 \cdot 3} {(y^{2})}^{3}}$

Multiply $2$ by $3$ .

$\frac{64 z^{3}}{27 x^{6} {(y^{2})}^{3}}$

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

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

$\frac{64 z^{3}}{27 x^{6} y^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{64 z^{3}}{27 x^{6} y^{6}}$