Question

What is `sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)` in simplified form?

Answer 1

`sqrt(2x^3)*sqrt(6x^2)*sqrt(10x) = 2x^3sqrt(30)`

Explanation:

Note that if `a, b >= 0` then

`sqrt(a)sqrt(b) = sqrt(ab)`

By extension, we find if `a, b, c >= 0` then:

`sqrt(a)sqrt(b)sqrt(c) = sqrt(ab)sqrt(c) = sqrt(abc)`

Note also that if `a>=0` then:

`sqrt(a^2) = a`

In our example, we will assume `x >= 0` in order that all of the original square roots are well defined Real numbers.

So in our example:

`sqrt(2x^3)*sqrt(6x^2)*sqrt(10x) = sqrt(2x^3*6x^2*10x)`

`color(white)(sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)) = sqrt(2x^3*2x^3*30)`

`color(white)(sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)) = sqrt((2x^3)^2)sqrt(30)`

`color(white)(sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)) = 2x^3 sqrt(30)`

Footnote

It seems to be a common error to assume that:

`sqrt(x^2) = x`

This does hold, but only if `x >= 0`.

If we want to cover the case `x < 0` too then we could write:

`sqrt(x^2) = abs(x)`

In the given example, we can deduce that the case `x >= 0` is intended, since otherwise `sqrt(2x^3)` and `sqrt(10x)` could take imaginary values.