Radicals in simplest form

A radical is said to be in simplest form when the root index (n) and the exponent of the radicand (m) have no common factors greater than 1, that is, when they are relatively prime (coprime): $$ \sqrt[n]{a^m} $$

A radical in simplest form cannot be expressed as an equivalent radical with a lower index.

A practical example

This radical is already in simplest form because the index n=3 and the exponent m=2 share no common factors.

$$ \sqrt[3]{a^2} $$

The integers 2 and 3 are relatively prime.

Note. The expression relatively prime (coprime) does not mean “prime numbers.” Two integers are coprime if they have no common factors, and they need not be prime themselves. For instance, 15 and 8 are coprime, even though neither is a prime number.

Example 2

This radical is not in simplest form because the index n=6 and the exponent m=4 have a common factor k=2.

$$ \sqrt[6]{a^4} $$

We simplify the radical by dividing both the index and the exponent by k=2.

$$ \sqrt[\frac{6}{2}]{a^{\frac{4}{2}}} $$

$$ \sqrt[3]{a^2} $$

The resulting expression is equivalent to the original radical.

Note. The simplified radical is now in simplest form because k=2 is the greatest common divisor of n=6 and m=4.

And so on.