Simplify the Radical Expression (x^(1/2))/(x^(-2/3))

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

Step 1

Move $x^{- \frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

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

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

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $2$ .

$x^{\frac{3 + 4}{6}}$

Add $3$ and $4$ .

$x^{\frac{7}{6}}$