Question

What is `3 sqrt(x^2/y)` in exponential form?

Answer 1

`(3x)/y^(1/2)`

Explanation:

`sqrt(x^2)` can be written as `x`, but y does not have a square root.

`3xy^(-1/2)`, but it is better not to have negative exponents.

`(3x)/y^(1/2)`

If you wanted to rationalize the denominator:

`(3x)/y^(1/2) xx y^(1/2)/y^(1/2)`

`(3xy^(1/2))/y`

Answer 2

Exponent form: `+-3xy^(-1/2)`

Explanation:

Here, `y>0` for `3sqrt(x^2/y)` to be real.
The form with exponents is `3((x^2)^(1/2))/y^(1/2)=3x/y^(1/2)`

`=3xy^(-1/2)`

Answer 3

`3sqrt(x^2/y) = 3abs(x)y^(-1/2)` for `y in (0,oo)`

Explanation:

Considering only Real valued square roots, we require:

`x^2/y >= 0`

Since `x^2>=0` for any Real `x`, this amounts to `y > 0`.

Note that `sqrt(x^2) = abs(x)` for any Real `x`. The square root sign denotes the principal square root, which in the case of Real square roots is the non-negative one.

Note that if `a >= 0` and `b > 0` then `sqrt(a/b) = sqrt(a)/sqrt(b)`

So we find:

`3sqrt(x^2/y) = 3(sqrt(x^2))/(sqrt(y)) = 3abs(x)/y^(1/2) = 3abs(x)y^(-1/2)`