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• Power Rules

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.

Ashok Kumar, B.E.

Founder of Math Doubts

A Specialist in Mathematics, Physics, and Engineering with 14 years of experience helping students master complex concepts from basics to advanced levels with clarity and precision.

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