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

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

Step 1

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

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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

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

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

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

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

Combine $3$ and $\frac{1}{2}$ .

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

Step 4

Cancel the common factor of $y^{\frac{3}{2}}$ .

Factor $y^{\frac{3}{2}}$ out of $x^{\frac{1}{3}} y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Add $3$ and $2$ .

$x^{\frac{5}{6}} y^{\frac{1}{2}}$

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

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