Simplify ((6x^-2y^5)/(2y^-3z^-11))^-3

${(\frac{6 x^{- 2} y^{5}}{2 y^{- 3} z^{- 11}})}^{- 3}$

Step 1

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

${(\frac{6 y^{5}}{2 y^{- 3} z^{- 11} x^{2}})}^{- 3}$

Step 2

Move $y^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{6 y^{5} y^{3}}{2 z^{- 11} x^{2}})}^{- 3}$

Step 3

Move $z^{- 11}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{6 y^{5} y^{3} z^{11}}{2 x^{2}})}^{- 3}$

Step 4

Multiply $y^{5}$ by $y^{3}$ by adding the exponents.

Move $y^{3}$ .

${(\frac{6 (y^{3} y^{5}) z^{11}}{2 x^{2}})}^{- 3}$

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

${(\frac{6 y^{3 + 5} z^{11}}{2 x^{2}})}^{- 3}$

Add $3$ and $5$ .

${(\frac{6 y^{8} z^{11}}{2 x^{2}})}^{- 3}$

Step 5

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6 y^{8} z^{11}$ .

${(\frac{2 (3 y^{8} z^{11})}{2 x^{2}})}^{- 3}$

Cancel the common factors.

Factor $2$ out of $2 x^{2}$ .

${(\frac{2 (3 y^{8} z^{11})}{2 (x^{2})})}^{- 3}$

Cancel the common factor.

${(\frac{2 (3 y^{8} z^{11})}{2 x^{2}})}^{- 3}$

Rewrite the expression.

${(\frac{3 y^{8} z^{11}}{x^{2}})}^{- 3}$

Step 6

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

${(\frac{x^{2}}{3 y^{8} z^{11}})}^{3}$

Step 7

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

Apply the product rule to $\frac{x^{2}}{3 y^{8} z^{11}}$ .

$\frac{{(x^{2})}^{3}}{{(3 y^{8} z^{11})}^{3}}$

Apply the product rule to $3 y^{8} z^{11}$ .

$\frac{{(x^{2})}^{3}}{{(3 y^{8})}^{3} {(z^{11})}^{3}}$

Apply the product rule to $3 y^{8}$ .

$\frac{{(x^{2})}^{3}}{3^{3} {(y^{8})}^{3} {(z^{11})}^{3}}$

Step 8

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

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

$\frac{x^{2 \cdot 3}}{3^{3} {(y^{8})}^{3} {(z^{11})}^{3}}$

Multiply $2$ by $3$ .

$\frac{x^{6}}{3^{3} {(y^{8})}^{3} {(z^{11})}^{3}}$

Step 9

Simplify the denominator.

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

$\frac{x^{6}}{27 {(y^{8})}^{3} {(z^{11})}^{3}}$

Multiply the exponents in ${(y^{8})}^{3}$ .

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

$\frac{x^{6}}{27 y^{8 \cdot 3} {(z^{11})}^{3}}$

Multiply $8$ by $3$ .

$\frac{x^{6}}{27 y^{24} {(z^{11})}^{3}}$

Multiply the exponents in ${(z^{11})}^{3}$ .

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

$\frac{x^{6}}{27 y^{24} z^{11 \cdot 3}}$

Multiply $11$ by $3$ .

$\frac{x^{6}}{27 y^{24} z^{33}}$

Simplify ((6x^-2y^5)/(2y^-3z^-11))^-3

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