Power Law of Radical Exponents

Formula

$\large b^{\frac{1}{n}} \,=\, \sqrt[\displaystyle n]{b}$

Proof

The rational numbers $\dfrac{1}{2}$, $\dfrac{1}{3}$, $\dfrac{1}{4}$ and etc. are denoted as square root ($\sqrt{\,\,\,\,}$), cube root ($\sqrt[\displaystyle 3]{\,\,\,\,}$), fourth root ($\sqrt[\displaystyle 4]{\,\,\,\,}$) and etc. respectively when the fractions are used as exponents of any number. The fractions become roots of the numbers.

Observe the following examples.

$(1) \,\,\,\,\,\,$ $6^{\frac{1}{2}} \,=\, \sqrt{6}$

$(2) \,\,\,\,\,\,$ $2^{\frac{1}{3}} \,=\, \sqrt[\displaystyle 3]{2}$

$(3) \,\,\,\,\,\,$ $7^{\frac{1}{4}} \,=\, \sqrt[\displaystyle 4]{7}$

$(4) \,\,\,\,\,\,$ $11^{\frac{1}{5}} \,=\, \sqrt[\displaystyle 5]{11}$

$(5) \,\,\,\,\,\,$ $9^{\frac{1}{6}} \,=\, \sqrt[\displaystyle 6]{9}$

$\,\,\,\,\,\, \vdots$

In the same way, if $b$ is a number and the $n$-th root of this literal number is written as $b^{\frac{1}{n}}$ or $\sqrt[\displaystyle n]{b}$. Therefore, the relation between them can be expressed in general form in algebraic form.

$\therefore \,\,\,\,\,\,$ $\large b^{\frac{1}{n}} \,=\, \sqrt[\displaystyle n]{b}$

The property of the exponents is called the power rule of radical exponents or indices.


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