How to Write the Square Root of in Simplest Radical Form

To write a square root in simplest radical form, you factor the number under the radical sign, pull out every factor that appears twice as a single number in front of the radical, and leave the rest inside. The result is an equivalent expression with the smallest possible number under the radical — for example, √72 = 6√2. The symbol “√” is called a radical, the number beneath it is the radicand, and a square root is simply a root of degree 2, the inverse of squaring a number (Wikipedia).

What Simplest Radical Form Means

A radical is in simplest form when the radicand has no factor that is a perfect square (other than 1). A perfect square is a number like 1, 4, 9, 16, 25, or 36 that results from multiplying an integer by itself. If any such factor is still sitting under the radical, the expression is not fully simplified.

The goal is always the same: strip out perfect-square factors and move their square roots to the front. For instance, √50 is not simplified because 25 — a perfect square — divides 50. Rewriting it as 5√2 brings it to simplest form, and the two expressions are equal in value.

Simplification does not change the number’s value; it only changes how the number is written. A simplified radical is usually preferred because it is easier to compare, combine, and compute with by hand. When a radicand has no perfect-square factor at all — such as √7 or √11 — it is already in simplest form and cannot be reduced further.

Find Perfect Square Factors

The reliable method is to factor the radicand completely and then group matching pairs. Each pair of identical factors produces one copy of that factor outside the radical; anything left unmatched stays inside.

Use the product rule

The product rule for radicals states that √(a × b) = √a × √b. This lets you split a large radicand into a perfect-square part and a leftover part, then simplify only the first. So √72 becomes √36 × √2, which equals 6√2.

Pull out the largest perfect square

You can remove perfect squares one at a time or find the largest one immediately — both routes reach the same answer. Pulling out the largest available perfect square in a single step is faster and reduces the chance of stopping too early.

Here are common perfect squares worth recognizing on sight:

• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
• Squares of larger integers: 169, 196, 225, 256, 289, 324, 361, 400

Knowing these by memory lets you spot factors quickly instead of testing every number.

A Step-by-Step Example

Simplify √72.

• Factor 72 as 36 × 2.
• Recognize 36 as a perfect square (6²).
• Apply the product rule: √72 = √36 × √2.
• Simplify: √36 = 6, so the result is 6√2.

Check: (6)² × 2 = 36 × 2 = 72, confirming the value is unchanged.

This same method covers related cases. If no factor repeats — as with √30, whose factors are 2 × 3 × 5 — then no perfect square divides the radicand except 1, and the radical is already in simplest form. And for a radicand that factors into several pairs, like 180 = 2 × 2 × 3 × 3 × 5, you pull out one 2 and one 3 to get 6√5.

A quick reference table

Practicing with a set like this builds the speed to simplify radicals mentally (Purplemath).

Common Mistakes to Avoid

• Leaving a perfect square in the radicand. Writing √8 as 2√2 is correct, but stopping at √8 is not. Always check that nothing remains that can be pulled out.
• Breaking the product rule. √(a + b) is not equal to √a + √b. The rule applies only to multiplication, so never split a sum under a radical.
• Simplifying a sum incorrectly. Expressions like √2 + √8 can be combined only after simplifying each term — √8 becomes 2√2, giving 3√2 — but √2 + √3 has no further simplification.
• Ignoring the answer’s sign. By convention, √x for a positive real number x denotes the positive square root only (Khan Academy). Keep that convention in mind when stating final answers.

Working carefully through the factor pairs catches most of these errors before they reach your final answer.

Key Takeaways

• Simplest radical form means the radicand contains no perfect-square factor other than 1.
• To simplify, find the largest perfect square that divides the radicand, take its square root outside the radical, and keep the remaining factor inside.
• The product rule √(a × b) = √a × √b is the tool that makes the split legal.
• If no perfect-square factor exists, as with √30, the radical is already simplified.
• Simplifying changes the look of the expression, never its value — verify with a square-and-multiply check.
• Avoid splitting sums under a radical and always confirm no perfect square remains.