Simplify (3x^-2)^4(x^2y^4)^3

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

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Combine $3$ and $\frac{1}{x^{2}}$ .

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

Step 3

Simplify the expression.

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

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

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

$\frac{81}{{(x^{2})}^{4}} {(x^{2} y^{4})}^{3}$

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

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

$\frac{81}{x^{2 \cdot 4}} {(x^{2} y^{4})}^{3}$

Multiply $2$ by $4$ .

$\frac{81}{x^{8}} {(x^{2} y^{4})}^{3}$

Apply the product rule to $x^{2} y^{4}$ .

$\frac{81}{x^{8}} ({(x^{2})}^{3} {(y^{4})}^{3})$

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

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

$\frac{81}{x^{8}} (x^{2 \cdot 3} {(y^{4})}^{3})$

Multiply $2$ by $3$ .

$\frac{81}{x^{8}} (x^{6} {(y^{4})}^{3})$

Step 4

Cancel the common factor of $x^{6}$ .

Factor $x^{6}$ out of $x^{8}$ .

$\frac{81}{x^{6} x^{2}} (x^{6} {(y^{4})}^{3})$

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

$\frac{81}{x^{6} x^{2}} (x^{6} ({(y^{4})}^{3}))$

Cancel the common factor.

$\frac{81}{x^{6} x^{2}} (x^{6} {(y^{4})}^{3})$

Rewrite the expression.

$\frac{81}{x^{2}} {(y^{4})}^{3}$

Step 5

Combine $\frac{81}{x^{2}}$ and ${(y^{4})}^{3}$ .

$\frac{81 {(y^{4})}^{3}}{x^{2}}$

Step 6

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

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

$\frac{81 y^{4 \cdot 3}}{x^{2}}$

Multiply $4$ by $3$ .

$\frac{81 y^{12}}{x^{2}}$