# How do you simplify (x^(3/2))/(3/2)?

Feb 16, 2016

${x}^{\frac{3}{2}} / \left(\frac{3}{2}\right) = \frac{2 {x}^{\frac{3}{2}}}{3} = \frac{2 \sqrt{{x}^{3}}}{3} = \frac{2 x \sqrt{x}}{3}$

#### Explanation:

The most important thing to know here is that dividing by a fraction is equivalent to multiplying by the fraction's reciprocal.

The expression we have here can be written as:

$= {x}^{\frac{3}{2}} \div \frac{3}{2}$

Instead of dividing by the fraction $\text{3/2}$, we can instead multiply by its reciprocal, $\text{2/3}$.

$= {x}^{\frac{3}{2}} \times \frac{2}{3}$

This can be written as

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

This is a fine simplification. However, if you want to simplify the fractional exponent, we can use the rule which states that

${x}^{\frac{a}{b}} = \sqrt[b]{{x}^{a}}$

Thus, the expression equals

$= \frac{2 \sqrt[2]{{x}^{3}}}{3} = \frac{2 \sqrt{{x}^{3}}}{3}$

We could simplify $\sqrt{{x}^{3}}$ by saying that $\sqrt{{x}^{3}} = \sqrt{{x}^{2}} \sqrt{x} = x \sqrt{x}$.

$= \frac{2 x \sqrt{x}}{3}$

This really becomes a matter of opinion as to where you wish to stop simplifying.