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

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

Step 1

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

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

Step 2

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

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

Step 3

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

Move $x^{- \frac{1}{3}}$ .

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

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

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Add $- 2$ and $3$ .

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

Step 4

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

$x^{\frac{1}{6}} y^{\frac{- 3 + 4}{6}}$

Add $- 3$ and $4$ .

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

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

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