Simplify ((3x^4y^6)/(4x^-1y^3))^2

${(\frac{3 x^{4} y^{6}}{4 x^{- 1} y^{3}})}^{2}$

Move $x^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

${(\frac{3 (x^{1} x^{4}) y^{6}}{4 y^{3}})}^{2}$

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

${(\frac{3 x^{1 + 4} y^{6}}{4 y^{3}})}^{2}$

Add $1$ and $4$ .

${(\frac{3 x^{5} y^{6}}{4 y^{3}})}^{2}$

Cancel the common factor of $y^{6}$ and $y^{3}$ .

Factor $y^{3}$ out of $3 x^{5} y^{6}$ .

${(\frac{y^{3} (3 x^{5} y^{3})}{4 y^{3}})}^{2}$

Cancel the common factors.

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

${(\frac{y^{3} (3 x^{5} y^{3})}{y^{3} \cdot 4})}^{2}$

Cancel the common factor.

${(\frac{y^{3} (3 x^{5} y^{3})}{y^{3} \cdot 4})}^{2}$

Rewrite the expression.

${(\frac{3 x^{5} y^{3}}{4})}^{2}$

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

Apply the product rule to $\frac{3 x^{5} y^{3}}{4}$ .

$\frac{{(3 x^{5} y^{3})}^{2}}{4^{2}}$

Apply the product rule to $3 x^{5} y^{3}$ .

$\frac{{(3 x^{5})}^{2} {(y^{3})}^{2}}{4^{2}}$

Apply the product rule to $3 x^{5}$ .

$\frac{3^{2} {(x^{5})}^{2} {(y^{3})}^{2}}{4^{2}}$

Simplify the numerator.

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

$\frac{9 {(x^{5})}^{2} {(y^{3})}^{2}}{4^{2}}$

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$\frac{9 x^{5 \cdot 2} {(y^{3})}^{2}}{4^{2}}$

Multiply $5$ by $2$ .

$\frac{9 x^{10} {(y^{3})}^{2}}{4^{2}}$

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

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

$\frac{9 x^{10} y^{3 \cdot 2}}{4^{2}}$

Multiply $3$ by $2$ .

$\frac{9 x^{10} y^{6}}{4^{2}}$

Raise $4$ to the power of $2$ .

$\frac{9 x^{10} y^{6}}{16}$