Math Doubts

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.



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more