Simplify Radicals Calculator factors a square root into a perfect-square coefficient and a leftover radicand, writing the result in the form a√b. Enter any nonnegative integer under the radical, and the tool pulls out every perfect-square factor it can find.
Factor out perfect squares
A square root simplifies whenever the number under it has a perfect-square factor greater than 1. The square root of 50 factors as the square root of (25 × 2), and since 25 is a perfect square (5^2), this becomes 5 times the square root of 2, written 5√2.
Simplify Radicals Calculator finds the largest such perfect-square factor automatically rather than requiring several rounds of partial factoring.
Work through the simplification of 72
Take the square root of 72. Factor 72 as 36 × 2, where 36 is a perfect square (6^2). The square root of 72 becomes 6 times the square root of 2, or 6√2. Checking this decimal: 6 × 1.41421… equals approximately 8.485, which matches the direct square root of 72 to the same precision.
Simplify Radicals Calculator shows both the perfect-square factor pulled out and the resulting decimal for this cross-check.
Recognize when a radical is already simplified
A square root is already in simplest form when the radicand has no perfect-square factor other than 1. The square root of 15 cannot be simplified further, since 15 factors only as 3 × 5, neither of which is a perfect square. Simplify Radicals Calculator returns √15 unchanged in this case, rather than forcing an unnecessary transformation.
Simplify a radical that is itself a perfect square
When the entire number under the root is a perfect square, the simplification collapses to a plain integer with no radical left over. The square root of 49 simplifies completely to 7, since 49 = 7^2 exactly, leaving no leftover radicand at all. Simplify Radicals Calculator displays this case as a bare integer rather than an awkward "7√1."
Use simplified radicals in further calculations
Simplified radical form matters most when a result needs to be combined with other radical terms, since only matching radicands can be added or subtracted directly. 3√2 + 5√2 combines to 8√2 because both terms share the same √2 factor, but 3√2 + 5√3 cannot combine further, since √2 and √3 are different radicands.
Simplifying each radical to its smallest possible radicand first is usually the step that reveals whether two radical terms can be combined at all.
Avoid this common mistake
A common error stops after pulling out one perfect-square factor instead of finding the largest one. Simplifying √72 as √(4 × 18) gives 2√18 first, which is correct but incomplete, since √18 still has a perfect-square factor of 9 inside it.
Continuing the simplification gives 2 × 3√2 = 6√2, matching the fully reduced form. Always check whether the leftover radicand still contains a perfect-square factor before stopping.
Simplify a radical using prime factorization
Prime factorization offers a systematic alternative to guessing perfect-square factors. Breaking 180 into primes gives 2 × 2 × 3 × 3 × 5. Grouping the primes in pairs identifies the perfect-square part: (2 × 2) and (3 × 3) each form a pair, leaving the unpaired 5 under the radical.
Each pair contributes one factor outside the root: 2 from the pair of 2s and 3 from the pair of 3s, multiplying to 6, so the square root of 180 simplifies to 6√5. Simplify Radicals Calculator uses this same pairing logic internally, which guarantees the largest perfect-square factor is found in a single pass.
Apply simplified radicals to the Pythagorean theorem
Simplified radical form appears naturally when computing a hypotenuse or leg length using the Pythagorean theorem, since the result of a sum of squares is rarely a perfect square itself.
A right triangle with legs of 6 and 8 has a hypotenuse of the square root of (36 + 64), which is the square root of 100, simplifying cleanly to 10.
A right triangle with legs of 4 and 6 instead gives the square root of (16 + 36), the square root of 52, which simplifies to 2√13, since 52 factors as 4 times 13 and 4 is a perfect square.
Frequently asked questions
How do you simplify a square root?
To simplify a square root, factor the number under the radical into a perfect square times a remaining factor, then move the square root of the perfect-square part outside the radical sign as a coefficient.
What is the simplified form of the square root of 72?
The simplified form of the square root of 72 is 6√2, since 72 factors as 36 × 2 and the square root of 36 is 6, leaving √2 under the radical.
How do you know if a radical is already simplified?
A radical is already simplified when the number under the root has no perfect-square factor greater than 1. Check the prime factorization of the radicand: if every prime factor appears to a power less than 2, the radical cannot be reduced further.
Can you add two radicals together?
Two radicals can be added together directly only when they share the same radicand after simplification, similar to combining like terms. 2√3 and 5√3 combine to 7√3, but 2√3 and 5√5 cannot combine into a single radical term.
What is the square root of a perfect square?
The square root of a perfect square is always a whole number with no radical remaining, since a perfect square is defined as a number that is some integer multiplied by itself. The square root of 100 is 10.
Why do you use the largest perfect-square factor when simplifying?
Using the largest perfect-square factor when simplifying avoids needing a second simplification pass, since a smaller factor leaves a leftover radicand that may still contain another perfect-square factor inside it.
How do you simplify the square root of 180 using prime factorization?
To simplify the square root of 180 using prime factorization, break it into 2 × 2 × 3 × 3 × 5, pair up the matching primes (2 and 2, then 3 and 3), pull one factor from each pair outside the radical (giving 2 × 3 = 6), and leave the unpaired 5 inside, giving 6√5.
Why do simplified radicals show up in Pythagorean theorem problems?
Simplified radicals show up in Pythagorean theorem problems because the sum of two squared leg lengths is rarely a perfect square itself, so the hypotenuse often comes out as a radical that can be reduced, such as √52 simplifying to 2√13.
Summary
Simplify Radicals Calculator writes a square root in the form a√b by pulling out the largest perfect-square factor from the radicand, collapsing to a plain integer when the entire input is a perfect square, and leaving the radical untouched when no perfect-square factor exists. Enter any nonnegative integer to see its fully reduced radical form alongside a decimal cross-check.