Simplify (27x^6y^-3)^(-2/3)

${(27 x^{6} y^{- 3})}^{- \frac{2}{3}}$

Step 1

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

${(27 x^{6} \frac{1}{y^{3}})}^{- \frac{2}{3}}$

Step 2

Multiply $27 x^{6} \frac{1}{y^{3}}$ .

Combine $27$ and $\frac{1}{y^{3}}$ .

${(x^{6} \frac{27}{y^{3}})}^{- \frac{2}{3}}$

Combine $x^{6}$ and $\frac{27}{y^{3}}$ .

${(\frac{x^{6} \cdot 27}{y^{3}})}^{- \frac{2}{3}}$

Step 3

Move $27$ to the left of $x^{6}$ .

${(\frac{27 \cdot x^{6}}{y^{3}})}^{- \frac{2}{3}}$

Step 4

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

${(\frac{y^{3}}{27 x^{6}})}^{\frac{2}{3}}$

Step 5

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

Apply the product rule to $\frac{y^{3}}{27 x^{6}}$ .

$\frac{{(y^{3})}^{\frac{2}{3}}}{{(27 x^{6})}^{\frac{2}{3}}}$

Apply the product rule to $27 x^{6}$ .

$\frac{{(y^{3})}^{\frac{2}{3}}}{27^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}$

Step 6

Multiply the exponents in ${(y^{3})}^{\frac{2}{3}}$ .

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

$\frac{y^{3 (\frac{2}{3})}}{27^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{y^{3 (\frac{2}{3})}}{27^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}$

Rewrite the expression.

$\frac{y^{2}}{27^{\frac{2}{3}} {(x^{6})}^{\frac{2}{3}}}$

Step 7

Simplify the denominator.

Rewrite $27$ as $3^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$\frac{y^{2}}{9 {(x^{6})}^{\frac{2}{3}}}$

Multiply the exponents in ${(x^{6})}^{\frac{2}{3}}$ .

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

$\frac{y^{2}}{9 x^{6 (\frac{2}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{y^{2}}{9 x^{3 (2) \frac{2}{3}}}$

Cancel the common factor.

$\frac{y^{2}}{9 x^{3 \cdot 2 \frac{2}{3}}}$

Rewrite the expression.

$\frac{y^{2}}{9 x^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{y^{2}}{9 x^{4}}$

Simplify (27x^6y^-3)^(-2/3)

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