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

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

Step 1

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

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

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

Combine the numerators over the common denominator.

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

Subtract $3$ from $1$ .

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

Divide $- 2$ by $2$ .

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

Step 2

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

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

Step 3

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

Multiply $x^{\frac{1}{3}}$ by $x$ .

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $3$ .

$x^{\frac{4}{3}}$