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

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

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

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

Simplify the numerator.

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

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

Multiply the exponents in ${(y^{3})}^{\frac{1}{6}}$ .

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

$\frac{y^{3 \frac{1}{3 \cdot 2}} x^{\frac{1}{6}} y^{\frac{1}{2}}}{x^{\frac{1}{3}}}$

Rewrite the expression.

$\frac{y^{\frac{1}{2}} x^{\frac{1}{6}} y^{\frac{1}{2}}}{x^{\frac{1}{3}}}$

Combine exponents.

Multiply $y^{\frac{1}{2}}$ by $y^{\frac{1}{2}}$ by adding the exponents.

Move $y^{\frac{1}{2}}$ .

$\frac{y^{\frac{1}{2}} y^{\frac{1}{2}} x^{\frac{1}{6}}}{x^{\frac{1}{3}}}$

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

$\frac{y^{\frac{1}{2} + \frac{1}{2}} x^{\frac{1}{6}}}{x^{\frac{1}{3}}}$

Combine the numerators over the common denominator.

$\frac{y^{\frac{1 + 1}{2}} x^{\frac{1}{6}}}{x^{\frac{1}{3}}}$

Add $1$ and $1$ .

$\frac{y^{\frac{2}{2}} x^{\frac{1}{6}}}{x^{\frac{1}{3}}}$

Divide $2$ by $2$ .

$\frac{y^{1} x^{\frac{1}{6}}}{x^{\frac{1}{3}}}$

Simplify $y^{1} x^{\frac{1}{6}}$ .

$\frac{y x^{\frac{1}{6}}}{x^{\frac{1}{3}}}$

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

$\frac{y}{x^{\frac{1}{3}} x^{- \frac{1}{6}}}$

Multiply $x^{\frac{1}{3}}$ by $x^{- \frac{1}{6}}$ by adding the exponents.

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

$\frac{y}{x^{\frac{1}{3} - \frac{1}{6}}}$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{x^{\frac{1}{3} \cdot \frac{2}{2} - \frac{1}{6}}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{y}{x^{\frac{2}{3 \cdot 2} - \frac{1}{6}}}$

Multiply $3$ by $2$ .

$\frac{y}{x^{\frac{2}{6} - \frac{1}{6}}}$

Combine the numerators over the common denominator.

$\frac{y}{x^{\frac{2 - 1}{6}}}$

Subtract $1$ from $2$ .

$\frac{y}{x^{\frac{1}{6}}}$

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

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