What is a rational exponent?

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Answer 1 · 2015-06-11T21:24:22

A rational exponent is an exponent of the form $\frac{m}{n}$ $m/n$ for two integers $m$ $m$ and $n$ $n$ , with the restriction $n \neq 0$ $n != 0$ .

$x^{\frac{m}{n}}$ $x^(m/n)$ is basically the same as $\sqrt[n]{x^{m}}$ $root(n)(x^m)$

Some general rules for exponents are:

$x^{0} = 1$ $x^0 = 1$

$x^{1} = x$ $x^1 = x$

$x^{- 1} = \frac{1}{x}$ $x^-1 = 1/x$

$x^{a} \cdot x^{b} = x^{a + b}$ $x^a * x^b = x^(a+b)$

${(x^{a})}^{b} = x^{a \cdot b}$ $(x^a)^b = x^(a*b)$

If $n$ $n$ is a positive integer then

$x^{\frac{1}{n}} = \sqrt[n]{x}$ $x^(1/n) = root(n)(x)$

From these rules, we can deduce:

${(\sqrt[n]{x})}^{m} = {(x^{\frac{1}{n}})}^{m} = x^{\frac{1}{n} \cdot m}$ $(root(n)(x))^m = (x^(1/n))^m = x^(1/n*m)$

$= x^{\frac{m}{n}}$ $=x^(m/n)$

$= x^{m \cdot \frac{1}{n}} = {(x^{m})}^{\frac{1}{n}} = \sqrt[n]{x^{m}}$ $=x^(m*1/n) = (x^m)^(1/n) = root(n)(x^m)$