Simplify ((5x^5y^3z^-2)/(10xy^5z^4))^-2

${(\frac{5 x^{5} y^{3} z^{- 2}}{10 x y^{5} z^{4}})}^{- 2}$

Step 1

Move $z^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{5 x^{5} y^{3}}{10 x y^{5} z^{4} z^{2}})}^{- 2}$

Step 2

Multiply $z^{4}$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

${(\frac{5 x^{5} y^{3}}{10 x y^{5} (z^{2} z^{4})})}^{- 2}$

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

${(\frac{5 x^{5} y^{3}}{10 x y^{5} z^{2 + 4}})}^{- 2}$

Add $2$ and $4$ .

${(\frac{5 x^{5} y^{3}}{10 x y^{5} z^{6}})}^{- 2}$

Step 3

Cancel the common factor of $5$ and $10$ .

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

${(\frac{5 (x^{5} y^{3})}{10 x y^{5} z^{6}})}^{- 2}$

Cancel the common factors.

Factor $5$ out of $10 x y^{5} z^{6}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Cancel the common factor of $x^{5}$ and $x$ .

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

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

Cancel the common factors.

Factor $x$ out of $2 x y^{5} z^{6}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $y^{3}$ and $y^{5}$ .

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

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

Cancel the common factors.

Factor $y^{3}$ out of $2 y^{5} z^{6}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

Step 7

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

Apply the product rule to $\frac{2 y^{2} z^{6}}{x^{4}}$ .

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

Apply the product rule to $2 y^{2} z^{6}$ .

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

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

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

Step 8

Simplify the numerator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $6$ by $2$ .

$\frac{4 y^{4} z^{12}}{{(x^{4})}^{2}}$

Step 9

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

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

$\frac{4 y^{4} z^{12}}{x^{4 \cdot 2}}$

Multiply $4$ by $2$ .

$\frac{4 y^{4} z^{12}}{x^{8}}$

Simplify ((5x^5y^3z^-2)/(10xy^5z^4))^-2

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