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Rational Power Rule of Exponents


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


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.


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

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

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

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

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


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

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