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

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

Step 1

Move ${(x^{- 2} y^{2})}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${({(3 x y^{- 5})}^{3} {(x^{- 2} y^{2})}^{4})}^{- 2}$

Step 2

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

${({(3 x \frac{1}{y^{5}})}^{3} {(x^{- 2} y^{2})}^{4})}^{- 2}$

Step 3

Multiply $3 x \frac{1}{y^{5}}$ .

Combine $3$ and $\frac{1}{y^{5}}$ .

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

Combine $x$ and $\frac{3}{y^{5}}$ .

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

Step 4

Move $3$ to the left of $x$ .

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

Step 5

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

Apply the product rule to $\frac{3 x}{y^{5}}$ .

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

Apply the product rule to $3 x$ .

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

Step 6

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

${(\frac{27 x^{3}}{{(y^{5})}^{3}} {(x^{- 2} y^{2})}^{4})}^{- 2}$

Step 7

Multiply the exponents in ${(y^{5})}^{3}$ .

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

${(\frac{27 x^{3}}{y^{5 \cdot 3}} {(x^{- 2} y^{2})}^{4})}^{- 2}$

Multiply $5$ by $3$ .

${(\frac{27 x^{3}}{y^{15}} {(x^{- 2} y^{2})}^{4})}^{- 2}$

Step 8

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

${(\frac{27 x^{3}}{y^{15}} {(\frac{1}{x^{2}} y^{2})}^{4})}^{- 2}$

Step 9

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

${(\frac{27 x^{3}}{y^{15}} {(\frac{y^{2}}{x^{2}})}^{4})}^{- 2}$

Step 10

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

${(\frac{27 x^{3}}{y^{15}} \cdot \frac{{(y^{2})}^{4}}{{(x^{2})}^{4}})}^{- 2}$

Step 11

Combine.

${(\frac{27 x^{3} {(y^{2})}^{4}}{y^{15} {(x^{2})}^{4}})}^{- 2}$

Step 12

Apply basic rules of exponents.

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

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

${(\frac{27 x^{3} y^{2 \cdot 4}}{y^{15} {(x^{2})}^{4}})}^{- 2}$

Multiply $2$ by $4$ .

${(\frac{27 x^{3} y^{8}}{y^{15} {(x^{2})}^{4}})}^{- 2}$

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

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

${(\frac{27 x^{3} y^{8}}{y^{15} x^{2 \cdot 4}})}^{- 2}$

Multiply $2$ by $4$ .

${(\frac{27 x^{3} y^{8}}{y^{15} x^{8}})}^{- 2}$

Step 13

Cancel the common factor of $x^{3}$ and $x^{8}$ .

Factor $x^{3}$ out of $27 x^{3} y^{8}$ .

${(\frac{x^{3} (27 y^{8})}{y^{15} x^{8}})}^{- 2}$

Cancel the common factors.

Factor $x^{3}$ out of $y^{15} x^{8}$ .

${(\frac{x^{3} (27 y^{8})}{x^{3} (y^{15} x^{5})})}^{- 2}$

Cancel the common factor.

${(\frac{x^{3} (27 y^{8})}{x^{3} (y^{15} x^{5})})}^{- 2}$

Rewrite the expression.

${(\frac{27 y^{8}}{y^{15} x^{5}})}^{- 2}$

Step 14

Cancel the common factor of $y^{8}$ and $y^{15}$ .

Factor $y^{8}$ out of $27 y^{8}$ .

${(\frac{y^{8} \cdot 27}{y^{15} x^{5}})}^{- 2}$

Cancel the common factors.

Factor $y^{8}$ out of $y^{15} x^{5}$ .

${(\frac{y^{8} \cdot 27}{y^{8} (y^{7} x^{5})})}^{- 2}$

Cancel the common factor.

${(\frac{y^{8} \cdot 27}{y^{8} (y^{7} x^{5})})}^{- 2}$

Rewrite the expression.

${(\frac{27}{y^{7} x^{5}})}^{- 2}$

Step 15

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

${(\frac{y^{7} x^{5}}{27})}^{2}$

Step 16

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

Apply the product rule to $\frac{y^{7} x^{5}}{27}$ .

$\frac{{(y^{7} x^{5})}^{2}}{27^{2}}$

Apply the product rule to $y^{7} x^{5}$ .

$\frac{{(y^{7})}^{2} {(x^{5})}^{2}}{27^{2}}$

Step 17

Simplify the numerator.

Multiply the exponents in ${(y^{7})}^{2}$ .

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

$\frac{y^{7 \cdot 2} {(x^{5})}^{2}}{27^{2}}$

Multiply $7$ by $2$ .

$\frac{y^{14} {(x^{5})}^{2}}{27^{2}}$

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$\frac{y^{14} x^{5 \cdot 2}}{27^{2}}$

Multiply $5$ by $2$ .

$\frac{y^{14} x^{10}}{27^{2}}$

Step 18

Raise $27$ to the power of $2$ .

$\frac{y^{14} x^{10}}{729}$

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

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