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

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

Step 1

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

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

Step 2

Multiply $x^{\frac{4}{3}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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

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

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

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

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

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

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

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

Multiply $2$ by $3$ .

$\frac{1}{- 3 x^{- \frac{3}{6} + \frac{4}{3} \cdot \frac{2}{2}}}$

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{1}{- 3 x^{- \frac{3}{6} + \frac{4 \cdot 2}{3 \cdot 2}}}$

Multiply $3$ by $2$ .

$\frac{1}{- 3 x^{- \frac{3}{6} + \frac{4 \cdot 2}{6}}}$

Combine the numerators over the common denominator.

$\frac{1}{- 3 x^{\frac{- 3 + 4 \cdot 2}{6}}}$

Simplify the numerator.

Multiply $4$ by $2$ .

$\frac{1}{- 3 x^{\frac{- 3 + 8}{6}}}$

Add $- 3$ and $8$ .

$\frac{1}{- 3 x^{\frac{5}{6}}}$

Step 3

Move the negative in front of the fraction.

$- \frac{1}{3 x^{\frac{5}{6}}}$