Simplify (4x^(2/3))/(5x^(1/2))

$\frac{4 x^{\frac{2}{3}}}{5 x^{\frac{1}{2}}}$

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

$\frac{4 x^{\frac{2}{3}} x^{- \frac{1}{2}}}{5}$

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

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

$\frac{4 (x^{- \frac{1}{2}} x^{\frac{2}{3}})}{5}$

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

$\frac{4 x^{- \frac{1}{2} + \frac{2}{3}}}{5}$

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

$\frac{4 x^{- \frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3}}}{5}$

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

$\frac{4 x^{- \frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{2}{2}}}{5}$

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

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

$\frac{4 x^{- \frac{3}{2 \cdot 3} + \frac{2}{3} \cdot \frac{2}{2}}}{5}$

Multiply $2$ by $3$ .

$\frac{4 x^{- \frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2}}}{5}$

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

$\frac{4 x^{- \frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2}}}{5}$

Multiply $3$ by $2$ .

$\frac{4 x^{- \frac{3}{6} + \frac{2 \cdot 2}{6}}}{5}$

Combine the numerators over the common denominator.

$\frac{4 x^{\frac{- 3 + 2 \cdot 2}{6}}}{5}$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{4 x^{\frac{- 3 + 4}{6}}}{5}$

Add $- 3$ and $4$ .

$\frac{4 x^{\frac{1}{6}}}{5}$