# How do you write 5x^(5/2) as a radical form?

Apr 23, 2016

$5 \sqrt{{x}^{5}}$ or $5 {x}^{2} \sqrt{x}$

#### Explanation:

If $x \ge 0$ then:

$5 {x}^{\frac{5}{2}} = 5 {x}^{5 \cdot \frac{1}{2}} = 5 {\left({x}^{5}\right)}^{\frac{1}{2}} = 5 \sqrt{{x}^{5}}$

If you prefer:

$5 {x}^{\frac{5}{2}} = 5 {x}^{2 + \frac{1}{2}} = 5 {x}^{2} {x}^{\frac{1}{2}} = 5 {x}^{2} \sqrt{x}$

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Footnote

Fractional exponents like this tend to work well with non-negative numbers, but can misbehave horribly for negative or Complex numbers.

For example:

${\left(- 1\right)}^{\frac{3}{2}} = - i \ne i = {\left({\left(- 1\right)}^{3}\right)}^{\frac{1}{2}}$

or even:

${\left({\left(- 1\right)}^{\frac{3}{2}}\right)}^{\frac{2}{3}} = {\left(- i\right)}^{\frac{2}{3}} = \frac{1}{2} - \frac{\sqrt{3}}{2} i \ne - 1 = {\left(- 1\right)}^{\frac{3}{2} \cdot \frac{2}{3}}$