Find the largest perfect-square factor, pull it out of the radical, and you have a simplified form. This guide shows exactly how to do that, step by step, with worked numbers and variables.
To simplify a square root, find the largest perfect-square factor of the number under the radical, take its square root, and write that result outside the radical sign with any remaining factor left inside. That one operation is the whole process.
A square root is in simplified form when the number under the radical, called the radicand, contains no perfect-square factor greater than 1. In other words, you cannot pull any whole number out of it. sqrt(50) is not simplified because 25 divides 50 evenly, and 25 is a perfect square. Once you write it as 5 sqrt(2), it is simplified, because 2 has no perfect-square factor above 1.
The rule that makes this possible is the product property of radicals, described on Wolfram MathWorld:
That property lets you split any radicand into a perfect-square piece and a leftover piece, then collapse the perfect-square piece into a coefficient.
You need these. Recognizing them on sight saves a lot of trial and error. The perfect squares from 1 through 225 are:
Each is just a whole number squared: 1² through 15². If you can spot that 144 divides 288, for instance, you jump straight to the answer instead of working through every factor one at a time.
Three steps cover every case involving a positive whole number.
If no perfect-square factor greater than 1 exists, the radical is already in simplest form. That is the entire method.
72 is a common exam number. It trips people up because they spot 4 first, pull out 2, and get 2 sqrt(18). That is not fully simplified, because 9 still divides 18. Always hunt for the largest perfect-square factor.
Want to check your work? Use the square root calculator to confirm the simplified radical form and decimal value for any number.
180 has more factors, so it gives more chances to miss the largest one. Let's go through it carefully.
If you had started with 4 instead of 36, you would have written 2 sqrt(45). Then you would need to simplify sqrt(45) = sqrt(9 x 5) = 3 sqrt(5), giving 2 x 3 sqrt(5) = 6 sqrt(5). Same answer, more steps. Starting with the largest factor is faster.
The product rule applies to variables the same way it applies to numbers. The key is the exponent. An even exponent under a radical simplifies cleanly; an odd one leaves a single power of the variable inside.
When the radicand mixes a number and a variable, handle both at the same time:
You can learn more about working with slopes and other algebraic expressions in the guide on how to find the slope of a line.
Not every radical can be reduced. If the radicand is a prime number, it has no perfect-square factor other than 1, so the radical stays as written. sqrt(7), sqrt(11), sqrt(13), sqrt(17), sqrt(19) are all already in simplest form.
The same is true when every prime factor of the radicand appears an odd number of times and no factor is repeated twice. sqrt(30) = sqrt(2 x 3 x 5) cannot be simplified, because 2, 3, and 5 each appear exactly once. There is nothing to pull out.
A quick test: factor the radicand into primes. If any prime appears at least twice, the radical simplifies. If no prime appears twice, it does not.
Enter any integer and see the simplified radical form, the factoring steps, and the decimal approximation.
Find the largest perfect-square factor of the radicand. Rewrite the radicand as that factor times the remainder. Apply the product rule to split the radical, then take the square root of the perfect-square factor and write it in front of the radical. The remainder stays under the radical sign. For example, sqrt(50) = sqrt(25 x 2) = 5 sqrt(2).
The perfect squares from 1 to 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Each is a whole number multiplied by itself: 1 through 10 squared. Knowing these by memory makes spotting factors much faster.
Not in the real number system. The square root of a negative number is imaginary. The imaginary unit i is defined as sqrt(-1), so sqrt(-9) = sqrt(9) x sqrt(-1) = 3i. This is covered in complex number arithmetic, not standard radical simplification.
Apply the product rule to the variable just as you would to a number. If the exponent under the radical is even, divide it by 2 and move the result outside: sqrt(x⁴) = x². If the exponent is odd, factor out the largest even power and leave one variable under the radical: sqrt(x⁵) = x² sqrt(x). Then simplify any numerical coefficient the same way.
Editor at Encore Editorial, Chris Terry sets the editorial standards here and turns dense topics into plain English. He has written widely on education, finance, and consumer markets.