Simplifying Radicals

To simplify a radical, divide both the index of the root (n) and the exponent of the radicand (m) by a common factor (k). $$ \sqrt[n]{a^m} = \sqrt[\frac{n}{k}]{a^{\frac{m}{k}}} $$

Simplifying a radical means rewriting it in a cleaner, more compact form without changing its value. The result is an equivalent expression with a smaller index, which is easier to read and work with.

To obtain an irreducible radical, divide the index and the exponent by their greatest common divisor.

$$ k = \mathrm{GCD}(n,m) $$

Note. A radical is irreducible when the index and the exponent have no common factors greater than 1. In other words, they are relatively prime (coprime). If common factors exist, the radical is reducible.

A practical example

Consider the radical

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

The greatest common divisor of the index n=6 and the exponent m=4 is k=2.

$$ \mathrm{GCD}(6,4) = 2 $$

Now divide both the index and the exponent by k=2.

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

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

The new radical is equivalent to the original, but written in simplest form.

Note. The expression is irreducible because the index n=3 and the exponent m=2 share no common factors greater than 1.

The underlying principle

This procedure follows directly from the invariance property of radicals.

$$ \sqrt[n]{a^m} = \sqrt[n \cdot p]{a^{m \cdot p}} = \sqrt[n : p]{a^{m : p}} $$

Multiplying or dividing both the index of the root and the exponent of the radicand by the same nonzero natural number produces an equivalent radical.

Equivalent radicals represent exactly the same numerical value.

Simplification using absolute value

If the radicand is negative (a<0) and raised to an even exponent (m even), it can be rewritten using the absolute value. $$ \sqrt[n]{a^m} = \sqrt[n]{|a|^m} $$

Example

Evaluate

$$ \sqrt[4]{(-3)^2} $$

A direct simplification of index and exponent would lead to a non-real expression.

$$ \sqrt[4]{(-3)^2} = \sqrt[2]{-3} = \text{not defined in } \mathbb{R} $$

Instead, observe that squaring a negative number always produces a nonnegative result.

$$ (-3)^2 = (+3)^2 = |-3|^2 $$

Rewrite the radicand using the absolute value.

$$ \sqrt[4]{|-3|^2} $$

Now simplify.

$$ \sqrt{|-3|} $$

Since |-3| = 3

$$ \sqrt{3} $$

Example 2

Evaluate

$$ \sqrt{(a-5)^2} $$

Note. Canceling the square root with the exponent without proper justification would give $$ \sqrt{(a-5)^2} = a-5 $$ which is incorrect. The expression a-5 may be negative, whereas an even-index root is always nonnegative. Therefore, the result is incorrect.

The correct reasoning is different.

The expression (a-5)² is always nonnegative.

Rewrite using the absolute value.

$$ \sqrt{|a-5|^2} $$

Simplify the radical.

$$ \sqrt{|a-5|^2} = |a-5| $$

Hence

$$ \sqrt{(a-5)^2} = |a-5| $$

Note. Because |a-5| is always nonnegative, it is the correct value of the radical.

And so on.