Simplify (((a^6)/(5b^-1))^2*(b/(9a^-11))^-1)^-3

${({(\frac{a^{6}}{5 b^{- 1}})}^{2} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

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

${({(\frac{a^{6} b}{5})}^{2} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

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

Apply the product rule to $\frac{a^{6} b}{5}$ .

${(\frac{{(a^{6} b)}^{2}}{5^{2}} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

Apply the product rule to $a^{6} b$ .

${(\frac{{(a^{6})}^{2} b^{2}}{5^{2}} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

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

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

${(\frac{a^{6 \cdot 2} b^{2}}{5^{2}} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

Multiply $6$ by $2$ .

${(\frac{a^{12} b^{2}}{5^{2}} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

Raise $5$ to the power of $2$ .

${(\frac{a^{12} b^{2}}{25} \cdot {(\frac{b}{9 a^{- 11}})}^{- 1})}^{- 3}$

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

${(\frac{a^{12} b^{2}}{25} \cdot {(\frac{b a^{11}}{9})}^{- 1})}^{- 3}$

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

${(\frac{a^{12} b^{2}}{25} \cdot \frac{9}{b a^{11}})}^{- 3}$

Combine.

${(\frac{a^{12} b^{2} \cdot 9}{25 (b a^{11})})}^{- 3}$

Cancel the common factor of $a^{12}$ and $a^{11}$ .

Factor $a^{11}$ out of $a^{12} b^{2} \cdot 9$ .

${(\frac{a^{11} (a b^{2} \cdot 9)}{25 (b a^{11})})}^{- 3}$

Cancel the common factors.

Factor $a^{11}$ out of $25 (b a^{11})$ .

${(\frac{a^{11} (a b^{2} \cdot 9)}{a^{11} (25 (b))})}^{- 3}$

Cancel the common factor.

${(\frac{a^{11} (a b^{2} \cdot 9)}{a^{11} (25 (b))})}^{- 3}$

Rewrite the expression.

${(\frac{a b^{2} \cdot 9}{25 (b)})}^{- 3}$

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

Factor $b$ out of $a b^{2} \cdot 9$ .

${(\frac{b (a b \cdot 9)}{25 (b)})}^{- 3}$

Cancel the common factors.

Factor $b$ out of $25 (b)$ .

${(\frac{b (a b \cdot 9)}{b \cdot 25})}^{- 3}$

Cancel the common factor.

${(\frac{b (a b \cdot 9)}{b \cdot 25})}^{- 3}$

Rewrite the expression.

${(\frac{a b \cdot 9}{25})}^{- 3}$

Move $9$ to the left of $a b$ .

${(\frac{9 \cdot (a b)}{25})}^{- 3}$

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

${(\frac{25}{9 a b})}^{3}$

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

Apply the product rule to $\frac{25}{9 a b}$ .

$\frac{25^{3}}{{(9 a b)}^{3}}$

Apply the product rule to $9 a b$ .

$\frac{25^{3}}{{(9 a)}^{3} b^{3}}$

Apply the product rule to $9 a$ .

$\frac{25^{3}}{9^{3} a^{3} b^{3}}$

Raise $25$ to the power of $3$ .

$\frac{15625}{9^{3} a^{3} b^{3}}$

Raise $9$ to the power of $3$ .

$\frac{15625}{729 a^{3} b^{3}}$

Simplify (((a^6)/(5b^-1))^2*(b/(9a^-11))^-1)^-3

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