Question

How do you know when a radical is in simplest form?

Answer 1

Hint: Simple radical has perfect square factors other than one. An expression that contains the ${n^{th}}$ root is known as the radical equation. The symbol of the radical is $\sqrt[n]{x}$. The radicand is a number that is in the radical.

Complete step by step solution:
Consider the expression as $\sqrt {36} $
The simplest form of the expression is\[\;6\] .
The radical expression is the square root of \[\;36\].
The radicand is 36.
Now consider the expression as $\sqrt[3]{{27}}$
The simplest form of the expression is $3$.
The radical expression is the cube root of \[\;27\].
The radicand is 27.
If the equation has ${n^{th}}$ root, then the simplest form has no factors of $n$.
 In short, we can create the criteria/conditions for which the radical can be recognized to be in its simplest form. Such that
A radical expression is in its simplest form when three conditions are met:
1. No radicands have perfect square factors other than 1
2. No radicand contains a fraction
3. No radicals appear in the denominator of a fraction.

A radical expression can be bought to simple form by
First prime factorization of it, then determining the index of the radical, after we have it done then we move the group of numbers from inside the radical or outside the radical. Then, we can simplify the expressions both inside and outside the radical by all the numbers inside the radical altogether.

Note: If the radical is somehow present in the denominator of the fraction, then we have multiplied the same radical with the terms with opposite sign in it such that the radical gets eliminated and comes in the form $(a - b)(a + b)$ and then can be easily simplified.