Simplify the radical when the radicand contains a fraction

Question

Radical: sqrt 5x^3/3

Radicand: 5x^3/3

Answer 1

Since we are not allowed to leave a fraction under a radical, we begin by splitting this into a fraction of two radicals, and then we rationalize the denominator. The original expression can be rewritten as √5x³ / √3. Now we will multiply both the numerator and denominator by √3, so the expression now becomes

(√5x³ · √3) / (√3 ·√3 ). Multiplying in the numerator, it simplifies to √15x³. Multiplying in the denominator, it simply becomes 3, so now the expression becomes √15x³/3.

There is one more step of simplifying to do. The x³ under the radical is big enough to take the square root of part of it, so if we break up the x³ as x² · x, we can take the square root of the x² part, bringing it out in front of the radical as "x" , and leave the other factor of x under the radical, so now the final form of the expression, simplified as much as possible, becomes x√15x/3.