# Why does sqrtx=x^(1/2)?

Feb 17, 2018

The reason this is true is that fractional exponents are defined that way.

For example, ${x}^{\frac{1}{2}}$ means the square root of $x$, and ${x}^{\frac{1}{3}}$ means the cube root of $x$. In general, ${x}^{\frac{1}{n}}$ means the $n$th root of $x$, written $\sqrt[n]{x}$.

You can prove it by using the law of exponents:

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

and

$\sqrt{x} \cdot \sqrt{x} = x$

Therefore, ${x}^{\frac{1}{2}} = \sqrt{x}$.