Solving Other Functions

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

Multiply the exponents, per the power of a power rule:

Define

Then if ,

it follows that

,

or, equivalently,

.

By the Intermediate Value Theorem (IVT), if is a continuous function, and and are of unlike sign, then for some . As a polynomial, is a continuous function, so the IVT applies here.

Evaluate for each of the following values: :

Only in the case of does it hold that assumes a different sign at both endpoints - . By the IVT, , and , for some .