Power with rational exponent and its properties

Power with a rational exponent

Let $a>0$ ,

m\in\mathbb{Z}

n\in\mathbb{N}

, $n\ge 2$ . Definition: $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\bigl(\sqrt[n]{a}\bigr)^m$ . Thus a rational exponent

\dfrac{m}{n}

is reduced to a root and an integer power. Special cases:

a^{\frac{1}{n}}=\sqrt[n]{a}

; for

m=1

you get exactly the root of degree

n

. Zero and negative exponent (for

a>0

): $a^0=1,\qquad a^{-r}=\dfrac{1}{a^r}\quad(r\in\mathbb{Q})$ .

In school,

a^r

for general rational

r

is usually studied for $a>0$ to avoid extra sign cases

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