Proof of Rational Power Rule of Exponents

The rational power rule of exponents can be derived in algebraic form to use it as a formula in mathematics.

Term with Rational Number as Power

$b$, $m$ and $n$ are three literals and they represent three different quantities. Take $b$ as a base and a fraction $\dfrac{m}{n}$ as exponent to form a special exponential term.

$b^{\Large \frac{m}{n}}$

Fractional Power in Product form

Now, write the fraction as product of two numbers to simplify the exponent.

$\implies$ $b^{\Large \frac{m}{n}} \,=\, b^{\, m \times \Large \frac{1}{n}}$

Simplifying the Expression

Now, use power rule of exponents to express the product of exponents as power of an exponent.

$\implies$ $b^{\Large \frac{m}{n}} \,=\, {\Big(b^m\Big)}^{\Large \frac{1}{n}} $

According to Radical power rule of exponents, the power of exponential term $b^m$ is a radical and it can be denoted by a radical symbol.

$\,\,\, \therefore \,\,\,\,\,\,$ $b^{\Large \frac{m}{n}} \,=\, \sqrt[\displaystyle n]{b^m}$