Simplify (-zy^-3)^2(4z^6y^9)^-2

${(- z y^{- 3})}^{2} {(4 z^{6} y^{9})}^{- 2}$

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

${(- z \frac{1}{y^{3}})}^{2} {(4 z^{6} y^{9})}^{- 2}$

Combine $\frac{1}{y^{3}}$ and $z$ .

${(- \frac{z}{y^{3}})}^{2} {(4 z^{6} y^{9})}^{- 2}$

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

Apply the product rule to $- \frac{z}{y^{3}}$ .

${(- 1)}^{2} {(\frac{z}{y^{3}})}^{2} {(4 z^{6} y^{9})}^{- 2}$

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

${(- 1)}^{2} \frac{z^{2}}{{(y^{3})}^{2}} {(4 z^{6} y^{9})}^{- 2}$

Raise $- 1$ to the power of $2$ .

$1 \frac{z^{2}}{{(y^{3})}^{2}} {(4 z^{6} y^{9})}^{- 2}$

Multiply $\frac{z^{2}}{{(y^{3})}^{2}}$ by $1$ .

$\frac{z^{2}}{{(y^{3})}^{2}} {(4 z^{6} y^{9})}^{- 2}$

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

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

$\frac{z^{2}}{y^{3 \cdot 2}} {(4 z^{6} y^{9})}^{- 2}$

Multiply $3$ by $2$ .

$\frac{z^{2}}{y^{6}} {(4 z^{6} y^{9})}^{- 2}$

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

$\frac{z^{2}}{y^{6}} \cdot \frac{1}{{(4 z^{6} y^{9})}^{2}}$

Combine fractions.

Combine.

$\frac{z^{2} \cdot 1}{y^{6} {(4 z^{6} y^{9})}^{2}}$

Multiply $z^{2}$ by $1$ .

$\frac{z^{2}}{y^{6} {(4 z^{6} y^{9})}^{2}}$

Simplify the denominator.

Apply the product rule to $4 z^{6} y^{9}$ .

$\frac{z^{2}}{y^{6} {(4 z^{6})}^{2} {(y^{9})}^{2}}$

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

$\frac{z^{2}}{y^{6} (4^{2} {(z^{6})}^{2}) {(y^{9})}^{2}}$

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

$\frac{z^{2}}{y^{6} (16 {(z^{6})}^{2}) {(y^{9})}^{2}}$

Multiply the exponents in ${(z^{6})}^{2}$ .

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

$\frac{z^{2}}{y^{6} (16 z^{6 \cdot 2}) {(y^{9})}^{2}}$

Multiply $6$ by $2$ .

$\frac{z^{2}}{y^{6} (16 z^{12}) {(y^{9})}^{2}}$

Multiply the exponents in ${(y^{9})}^{2}$ .

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

$\frac{z^{2}}{y^{6} (16 z^{12}) y^{9 \cdot 2}}$

Multiply $9$ by $2$ .

$\frac{z^{2}}{y^{6} (16 z^{12}) y^{18}}$

$\frac{z^{2}}{y^{6} \cdot 16 z^{12} y^{18}}$

Multiply $y^{6}$ by $y^{18}$ by adding the exponents.

Move $y^{18}$ .

$\frac{z^{2}}{y^{18} y^{6} \cdot 16 z^{12}}$

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

$\frac{z^{2}}{y^{18 + 6} \cdot 16 z^{12}}$

Add $18$ and $6$ .

$\frac{z^{2}}{y^{24} \cdot 16 z^{12}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $z^{2}$ and $z^{12}$ .

Multiply by $1$ .

$\frac{z^{2} \cdot 1}{y^{24} \cdot 16 z^{12}}$

Cancel the common factors.

Factor $z^{2}$ out of $y^{24} \cdot 16 z^{12}$ .

$\frac{z^{2} \cdot 1}{z^{2} (y^{24} \cdot 16 z^{10})}$

Cancel the common factor.

$\frac{z^{2} \cdot 1}{z^{2} (y^{24} \cdot 16 z^{10})}$

Rewrite the expression.

$\frac{1}{y^{24} \cdot 16 z^{10}}$

Move $16$ to the left of $y^{24}$ .

$\frac{1}{16 y^{24} z^{10}}$

Simplify (-zy^-3)^2(4z^6y^9)^-2

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