How to Simplify Square Roots and Other Radicals

To simplify a square root, factor the number under the radical into a perfect square times another number, then take the square root of the perfect square. For example, 72 = 36 × 2, so √72 = √36 × √2 = 6√2. A square root is fully simplified when the number left under the radical has no perfect-square factor other than 1.

The method rests on one property of radicals:

Simplified radicals are exact. 6√2 and 8.48528… describe the same number, but the radical form keeps full precision and makes like terms visible, which is why algebra and geometry answers are usually given that way.

Method 1: The largest perfect-square factor

1. List perfect squares up to the number: 4, 9, 16, 25, 36, 49, 64, 81, 100, …
2. Find the largest one that divides the number evenly.
3. Split the radical and take the square root of the perfect square.

If you use a smaller perfect square by mistake, you can keep going. √72 = √4 × √18 = 2√18, and √18 = √9 × √2 = 3√2, so 2√18 = 6√2. Same answer, one extra step.

The list of perfect squares continues the table.

Method 2: Prime factorization

For larger numbers, break the radicand into primes and pull out each pair. Every pair of identical primes leaves the radical as a single factor.

This method never misses a factor, so it is the safest choice when the perfect square is not obvious. The prime factorization calculator gives you the primes.

Common simplified square roots

Cube roots and higher roots

The same idea works for any index. For a cube root, look for perfect cubes (8, 27, 64, 125, …), or groups of three identical primes.

• ∛54 = ∛(27 × 2) = 3∛2
• ∛250 = ∛(125 × 2) = 5∛2
• ⁴√48 = ⁴√(16 × 3) = 2⁴√3

The cube root calculator and nth root calculator handle the decimal values.

Radicals with variables

Treat each variable’s exponent like a count of primes: every pair comes out. Assuming the variables are not negative:

If a variable could be negative, √(x²) equals |x|, not x, because a square root is never negative.

Operations with radicals

Adding and subtracting

Combine only like radicals, which have the same index and the same radicand. Simplify first, because unlike-looking radicals are often alike underneath.

Multiplying

Multiply the numbers outside together and the numbers inside together, then simplify.

Rationalizing the denominator

By convention, a simplified answer has no radical in the denominator. Multiply the top and bottom by whatever removes it.

• A single root: 5/√3 = (5 × √3)/(√3 × √3) = 5√3/3
• A binomial: multiply by the conjugate, which flips the sign between the terms. 4/(3 − √5) × (3 + √5)/(3 + √5) = 4(3 + √5)/(9 − 5) = 3 + √5

The conjugate works because (a − b)(a + b) = a² − b², which squares away the radical.

Estimating a square root without a calculator

Simplifying gives the exact form; sometimes you also need a decimal. Two quick hand methods get you there.

Bracket and adjust. Find the perfect squares on either side, then add the leftover divided by twice the lower root:

For √50: 49 is the nearest square below, so √50 ≈ 7 + 1/14 ≈ 7.0714. The true value is 7.0711, so the estimate is off by less than 0.001.

Divide and average. Pick a guess, divide the number by it, and average the guess with the result. Each round roughly doubles the number of correct digits. For √20, guess 4.5: 20 ÷ 4.5 = 4.444, and the average of 4.5 and 4.444 is 4.4722, already correct to four decimal places (the true value is 4.47214). This ancient technique, often called the Babylonian method, is essentially what calculators do internally.

Combine the methods with simplification for larger numbers: √72 = 6√2, and √2 ≈ 1.41421, so √72 ≈ 8.4853.

Common mistakes

• Splitting a sum. √(9 + 16) is √25 = 5, not √9 + √16 = 7. The product rule applies only to multiplication.
• Stopping at a non-largest square. 2√18 is correct but not simplified, since 18 still contains 9.
• Taking a root of the outside number. In 3√12, only the 12 is under the radical: 3√12 = 3 × 2√3 = 6√3.
• Forgetting the negative root when solving. √49 = 7, but the equation x² = 49 has two solutions, ±7. See how to solve quadratic equations.

Square roots also appear in fractional exponents, √x = x^1/2, so the exponent rules apply to radicals too. To check any answer, the simplify radicals calculator shows the factorization, and the square root calculator gives the decimal value.