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

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

Step 1

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

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

Step 2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

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

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

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

Add $3$ and $4$ .

${(\frac{2 y^{5}}{- 3 x^{7} y^{- 2}})}^{- 3}$

Step 3

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

${(\frac{2 y^{5} y^{2}}{- 3 x^{7}})}^{- 3}$

Step 4

Multiply $y^{5}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

${(\frac{2 (y^{2} y^{5})}{- 3 x^{7}})}^{- 3}$

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

${(\frac{2 y^{2 + 5}}{- 3 x^{7}})}^{- 3}$

Add $2$ and $5$ .

${(\frac{2 y^{7}}{- 3 x^{7}})}^{- 3}$

Step 5

Move the negative in front of the fraction.

${(- \frac{2 y^{7}}{(3) x^{7}})}^{- 3}$

Step 6

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

${(- \frac{3 x^{7}}{2 y^{7}})}^{3}$

Step 7

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

Apply the product rule to $- \frac{3 x^{7}}{2 y^{7}}$ .

${(- 1)}^{3} {(\frac{3 x^{7}}{2 y^{7}})}^{3}$

Apply the product rule to $\frac{3 x^{7}}{2 y^{7}}$ .

${(- 1)}^{3} \frac{{(3 x^{7})}^{3}}{{(2 y^{7})}^{3}}$

Apply the product rule to $3 x^{7}$ .

${(- 1)}^{3} \frac{3^{3} {(x^{7})}^{3}}{{(2 y^{7})}^{3}}$

Apply the product rule to $2 y^{7}$ .

${(- 1)}^{3} \frac{3^{3} {(x^{7})}^{3}}{2^{3} {(y^{7})}^{3}}$

Step 8

Raise $- 1$ to the power of $3$ .

$- \frac{3^{3} {(x^{7})}^{3}}{2^{3} {(y^{7})}^{3}}$

Step 9

Simplify the numerator.

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

$- \frac{27 {(x^{7})}^{3}}{2^{3} {(y^{7})}^{3}}$

Multiply the exponents in ${(x^{7})}^{3}$ .

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

$- \frac{27 x^{7 \cdot 3}}{2^{3} {(y^{7})}^{3}}$

Multiply $7$ by $3$ .

$- \frac{27 x^{21}}{2^{3} {(y^{7})}^{3}}$

Step 10

Simplify the denominator.

Raise $2$ to the power of $3$ .

$- \frac{27 x^{21}}{8 {(y^{7})}^{3}}$

Multiply the exponents in ${(y^{7})}^{3}$ .

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

$- \frac{27 x^{21}}{8 y^{7 \cdot 3}}$

Multiply $7$ by $3$ .

$- \frac{27 x^{21}}{8 y^{21}}$

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

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