Power Law of Power of Radical Exponents

Formula

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

Proof

$b$ is a literal number and assume its exponent is a radical number and it is in the form of $\dfrac{m}{n}$.

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

The rational number can be written as the product of two numbers.

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

The power rule of power of an exponential term tells that the product of the exponents can be written as the power of another exponent.

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

The exponent of the exponential term $b^{\displaystyle m}$ is a radical. So, it can be written as the root of the exponential term as per the power rule of radical exponents.

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

This property of the exponents is called the power rule of power of a radical exponents.

Verification

For example, $\large 2^{\frac{3}{4}}$ $\,=\,$ $1.681793$

Now, check it in alternative method.

$\large \sqrt[\displaystyle 4]{2^3}$ $\,=\,$ $\large \sqrt[\displaystyle 4]{8}$

$\implies$ $\large \sqrt[\displaystyle 4]{2^3}$ $\,=\,$ $1.681793$

Now compare both results and they both are equal exactly.

$\therefore \,\,\,\,\,\,$ $\large 2^{\frac{3}{4}}$ $\,=\,$ $\large \sqrt[\displaystyle 4]{2^3}$ $\,=\,$ $1.681793$

Hence, exponential identity is true for all values and it is verified mathematically.


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