Simplify ((16x^(1/6)y^-2)/(x^(-1/6)y^6))^(3/2)

${(\frac{16 x^{\frac{1}{6}} y^{- 2}}{x^{- \frac{1}{6}} y^{6}})}^{\frac{3}{2}}$

Step 1

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

${(\frac{16 x^{\frac{1}{6}}}{x^{- \frac{1}{6}} y^{6} y^{2}})}^{\frac{3}{2}}$

Step 2

Multiply $y^{6}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

${(\frac{16 x^{\frac{1}{6}}}{x^{- \frac{1}{6}} (y^{2} y^{6})})}^{\frac{3}{2}}$

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

${(\frac{16 x^{\frac{1}{6}}}{x^{- \frac{1}{6}} y^{2 + 6}})}^{\frac{3}{2}}$

Add $2$ and $6$ .

${(\frac{16 x^{\frac{1}{6}}}{x^{- \frac{1}{6}} y^{8}})}^{\frac{3}{2}}$

Step 3

Move $x^{- \frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{16 x^{\frac{1}{6}} x^{\frac{1}{6}}}{y^{8}})}^{\frac{3}{2}}$

Step 4

Multiply $x^{\frac{1}{6}}$ by $x^{\frac{1}{6}}$ by adding the exponents.

Move $x^{\frac{1}{6}}$ .

${(\frac{16 (x^{\frac{1}{6}} x^{\frac{1}{6}})}{y^{8}})}^{\frac{3}{2}}$

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

${(\frac{16 x^{\frac{1}{6} + \frac{1}{6}}}{y^{8}})}^{\frac{3}{2}}$

Combine the numerators over the common denominator.

${(\frac{16 x^{\frac{1 + 1}{6}}}{y^{8}})}^{\frac{3}{2}}$

Add $1$ and $1$ .

${(\frac{16 x^{\frac{2}{6}}}{y^{8}})}^{\frac{3}{2}}$

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

Factor $2$ out of $2$ .

${(\frac{16 x^{\frac{2 (1)}{6}}}{y^{8}})}^{\frac{3}{2}}$

Cancel the common factors.

Factor $2$ out of $6$ .

${(\frac{16 x^{\frac{2 \cdot 1}{2 \cdot 3}}}{y^{8}})}^{\frac{3}{2}}$

Cancel the common factor.

${(\frac{16 x^{\frac{2 \cdot 1}{2 \cdot 3}}}{y^{8}})}^{\frac{3}{2}}$

Rewrite the expression.

${(\frac{16 x^{\frac{1}{3}}}{y^{8}})}^{\frac{3}{2}}$

Step 5

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

Apply the product rule to $\frac{16 x^{\frac{1}{3}}}{y^{8}}$ .

$\frac{{(16 x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Apply the product rule to $16 x^{\frac{1}{3}}$ .

$\frac{16^{\frac{3}{2}} {(x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Step 6

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

$\frac{{(4^{2})}^{\frac{3}{2}} {(x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

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

$\frac{4^{2 (\frac{3}{2})} {(x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4^{2 (\frac{3}{2})} {(x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{4^{3} {(x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Raise $4$ to the power of $3$ .

$\frac{64 {(x^{\frac{1}{3}})}^{\frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Multiply the exponents in ${(x^{\frac{1}{3}})}^{\frac{3}{2}}$ .

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

$\frac{64 x^{\frac{1}{3} \cdot \frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{64 x^{\frac{1}{3} \cdot \frac{3}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{64 x^{\frac{1}{2}}}{{(y^{8})}^{\frac{3}{2}}}$

Step 7

Multiply the exponents in ${(y^{8})}^{\frac{3}{2}}$ .

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

$\frac{64 x^{\frac{1}{2}}}{y^{8 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$\frac{64 x^{\frac{1}{2}}}{y^{2 (4) \frac{3}{2}}}$

Cancel the common factor.

$\frac{64 x^{\frac{1}{2}}}{y^{2 \cdot 4 \frac{3}{2}}}$

Rewrite the expression.

$\frac{64 x^{\frac{1}{2}}}{y^{4 \cdot 3}}$

Multiply $4$ by $3$ .

$\frac{64 x^{\frac{1}{2}}}{y^{12}}$

Simplify ((16x^(1/6)y^-2)/(x^(-1/6)y^6))^(3/2)

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