When we define nth root, why is n<0 exlcuded? For example, why do we not define \sqrt[-2]{x}?

Question

When we define nth root,

why is n<0 exlcuded?

For example,
why do we not define
`\sqrt[-2]{x}`?

Answer 1

other notations work better

Explanation:

You want to know why do we not define roots with negative indices:

`\displaystyle \sqrt[{-2}]{x}`

Roots

The index in a surd represents the denominator of a fractional power:

`\sqrt[n]{x}=x^(1)/(n)`

Using a negative number for a root index is the same as inverting the root:

`\sqrt[-n]{x}=x^{-(1)/(n)}=\frac{1}{x^{(1)/(n)}}=\frac{1}{\sqrt[n]{x}}`

Essentially, there are more straightforward, less confusing and error-prone ways to write the same expression.

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Additional comment

Some calculators handle this easily. Others may not.

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