Question

How do I rationalize Cubic root?

Answer 1

Use the fact that `root(3)(a^3)=a`
Together with `root(3)a root(3)b = root(3)(ab)`
(And, of course, the fact that multiplying by one may change the way a number is written, but it does not change the (value of) the number.)
Example 1
Rationalize the denominator: `2/root(3)5`

We'll use the facts mentioned above to write:

`2/root(3)5=2/root(3)5 * root(3)(5^2)/root(3)(5^2) = (2root(3)25)/root(3)(5^3) = (2root(3)25)/5`

Example 2
Rationalize the denominator: `7/root(3)4` .
We could multiply by `root(3)(4^2)/root(3)(4^2)`, but `root(3)16` is reducible!

We'll take a more direct path to the solution if we Realize that what we have is:`7/root(3)(2^2)` so we only need to multiply by `root(3)(2)/root(3)(2)`,

`7/root(3)4 = 7/root(3)4 *root(3)(2)/root(3)(2) = (7root(3)2)/root(3)(2^3) = (7root(3)2)/2`

Example 3 (last)
If the denominator is `root(3)20`, the similar path to rationalizing would be:

`root(3)20 = root(3)(2^2*5)`, so we would multiply by

think . . . (what is missing to make a perfect cube?

think some more, you can get it. . .

.

.

multiply numerator and denominator by `root(3)(2*5^2)`.