Simplify (2x^2y^-3z^-1)^-2

${(2 x^{2} y^{- 3} z^{- 1})}^{- 2}$

Step 1

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

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

Step 2

Multiply $2 x^{2} \frac{1}{y^{3}}$ .

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

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

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

${(\frac{x^{2} \cdot 2}{y^{3}} z^{- 1})}^{- 2}$

Step 3

Move $2$ to the left of $x^{2}$ .

${(\frac{2 \cdot x^{2}}{y^{3}} z^{- 1})}^{- 2}$

Step 4

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

${(\frac{2 x^{2}}{y^{3}} \cdot \frac{1}{z})}^{- 2}$

Step 5

Combine.

${(\frac{2 x^{2} \cdot 1}{y^{3} z})}^{- 2}$

Step 6

Multiply $2$ by $1$ .

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

Step 7

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

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

Step 8

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

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

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

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

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

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

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

Step 9

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

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

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

Multiply $3$ by $2$ .

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

Step 10

Simplify the denominator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

Simplify (2x^2y^-3z^-1)^-2

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