Simplest Form of a Radical
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
The simplest form of a radical refers to a radical expression where the radicand (the number or expression under the radical) is as simplified as possible, and there are no perfect square factors (or higher power factors, depending on the type of root) left under the radical. In this form, the radical expression should not contain any factors that can be simplified further.
Key points for a radical to be in its simplest form:
No perfect square factors (or higher power factors) should remain inside the radical. This means you should factor out any perfect squares (or cubes, depending on the root) from under the radical.
No fractions inside the radical. If the radicand is a fraction, the numerator and denominator should each be simplified to the extent possible, and the denominator should not contain a radical.
No radical in the denominator. If the denominator contains a radical, it should be rationalized.
Example 1: Simplifying a square root
Simplifying sqrt(18):
Factor 18 as 18 = 9 • 2.
Since 9 is a perfect square, sqrt(18) = sqrt(9 • 2) = sqrt(9) • sqrt(2).
Simplify: sqrt(9) = 3, so sqrt(18) = 3sqrt(2).
Thus, 3sqrt(2) is the simplest form of sqrt(18).
Example 2: Rationalizing the denominator
Simplifying 1 / sqrt(5):
Multiply both the numerator and denominator by sqrt(5) to eliminate the radical in the denominator:
Now, the expression is in its simplest form, as the denominator is rationalized, and there are no further simplifications needed.
Conclusion:
The simplest form of a radical means:
The radicand is as simplified as possible (no perfect square factors remain inside).
There is no radical in the denominator.
The expression is fully simplified and in its most compact form.