Question

How can you use a rational exponent to
represent a power involving a radical?
​

Answer 1

`\sqrt[n]{x^m}=x^{(m)/(n)}`

Explanation:

A radical represents a fractional power. For example, ...

`√(x)=x^{(1)/(2)}`

This makes sense in view of the rules of exponents for multiplication.

`a^ba^c=a^(b+c)\\\\a^{(1)/(2)}a^{(1)/(2)}=a^{(1)/(2)+(1)/(2)}=a\\\\(√(a))(√(a))=a`

So, a root other than a square root can be similarly represented by a fractional exponent.

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The power of a radical and the radical of a power are the same thing. That is, it doesn't matter whether the power is outside or inside the radical.

`\sqrt[n]{x^m}=x^{(m)/(n)}=(\sqrt[n]{x})^m`