Rational Power Rule of Exponents

Formula

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

Introduction

The numbers contain rational numbers as exponents in some special cases. The value of each exponential term can be evaluated by calculating the square root or higher order root for power of the number.

Examples

$(1) \,\,\,\,\,\,$ $3^{\large \frac{1}{2}} \,=\, \sqrt{3}$

$(2) \,\,\,\,\,\,$ $5^{\large \frac{4}{3}} \,=\, \sqrt[\Large 3]{5^4}$

$(3) \,\,\,\,\,\,$ $2^{\large \frac{3}{7}} \,=\, \sqrt[\Large 7]{2^3}$

$(4) \,\,\,\,\,\,$ $11^{\large \frac{2}{5}} \,=\, \sqrt[\Large 5]{11^2}$

$(5) \,\,\,\,\,\,$ $9^{\large \frac{13}{6}} \,=\, \sqrt[\Large 6]{9^{13}}$

Proof

Learn how to derive the rational power rule of indices in algebraic form.