Evaluate (x^-2y^(1/2))/(x^(1/3)y^-1)

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

Step 1

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

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

Step 2

Multiply $x^{\frac{1}{3}}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

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

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

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

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

Combine $2$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

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

Add $6$ and $1$ .

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

Step 3

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

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

Step 4

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

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

Raise $y$ to the power of $1$ .

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

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

$\frac{y^{\frac{1}{2} + 1}}{x^{\frac{7}{3}}}$

Write $1$ as a fraction with a common denominator.

$\frac{y^{\frac{1}{2} + \frac{2}{2}}}{x^{\frac{7}{3}}}$

Combine the numerators over the common denominator.

$\frac{y^{\frac{1 + 2}{2}}}{x^{\frac{7}{3}}}$

Add $1$ and $2$ .

$\frac{y^{\frac{3}{2}}}{x^{\frac{7}{3}}}$