Question

How do you rationalize the denominator and simplify `root3( (1 / (2x^2)))`?

Answer 1

You `ul("force")` it into something you can take the cube root of.

`=root(3)(4x)/(2x)`

Explanation:

Multiply by 1 and you do not change the intrinsic value. However, 1 comes in many forms

`color(green)(root(3)(1/(2x^2)color(red)(xx1)))`

`2xx2^2=2^3`

`x^2xx x^2xx x^2 =x^2xx(x^2)^2 = (x^2)^3`

`color(green)(root(3)(1/(2x^2)color(red)(xx(2^2(x^2)^2)/(2^2(x^2)^2))))" "->" "root(3)((4x^4)/(2^3(x^2)^3))`

`color(white)("vvvvvvvvvvvvvvv")->" "root(3)(4x^4)/(2x^2)`
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Simplified further

`x^4->x^3xx x" "` so we can change the `4x^4` such that we have:

`(xroot(3)4x)/(2x^2) = x/x xxroot(3)(4x)/(2x)`

`=root(3)(4x)/(2x)`