Square Root Calculator

Enter a number and get its square root two ways: the decimal answer and the exact simplified radical form (so 72 gives you both 8.48528 and 6√2), with the prime factorization steps shown. Switch modes for cube roots or any nth root, and negative numbers are handled honestly with imaginary results.

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How this square root calculator works

Most calculators hand you a decimal and walk away. This one gives you the answer your math teacher actually wants: the exact simplified radical form, found by prime factorization, alongside the decimal rounded to 6 significant figures. Type 72 and you get both √72 ≈ 8.48528 and the exact form 6√2, with every step of the simplification shown using your number. It also spots perfect squares and perfect cubes instantly, and it answers honestly when you feed it a negative.

Switch to cube root mode for ∛x, or nth root mode for any root from 2 to 50. The same prime factorization logic applies: cube roots pull out triples of factors instead of pairs.

The formula

√x = x1/2    and in general    n√x = x1/n

Here x is your number (the radicand) and n is the root index: 2 for square roots, 3 for cube roots. A root is just a fractional exponent, which is why the exponent rules you already know carry over: √(a × b) = √a × √b. That single rule is what makes simplifying radicals possible.

How to simplify a square root (the prime factorization method)

This is the part homework actually asks for, so let's do it properly. A radical is simplified when nothing under the root sign has a square factor left in it. The reliable way to get there has three steps:

  1. Factor the number into primes. Keep dividing by the smallest prime that fits: 72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2 × 2 × 2 × 3 × 3.
  2. Group matching primes into pairs. One pair of 2s, one pair of 3s, and a lonely 2 left over: 72 = (2 × 3)² × 2.
  3. Pull one factor from each pair outside the root. The pairs come out as 2 × 3 = 6 and the leftover stays inside: √72 = 6√2.

Why pairs? Because √(p × p) = p. Every matched pair under a square root collapses to a single copy outside it. For cube roots you hunt for triples instead, and for an nth root you hunt for groups of n. The calculator above runs exactly this method on your number and shows you each step, so you can copy the working, not just the answer.

Same idea, one more example: √50 = √(5 × 5 × 2) = 5√2. And a trap worth knowing: √48 = √(2 × 2 × 2 × 2 × 3) has two pairs of 2s, so it is 4√3, not 2√12. If you stop simplifying too early, the thing left inside still has a square factor hiding in it. Finding the largest perfect square factor is really a greatest common factor kind of skill: it rewards knowing your factors cold.

Worked example

Simplify and evaluate √72.

Prime factorization: 72 = 2 × 2 × 2 × 3 × 3. Pair up matching primes: 72 = (2 × 3)² × 2. Take one factor from each pair outside the root: √72 = 6√2.

For the decimal, √2 = 1.41421, so √72 = 6 × 1.41421 = 8.48528. Sanity check: 72 sits between the perfect squares 64 (8²) and 81 (9²), so an answer between 8 and 9 is exactly what we should see.

Perfect squares from 1 to 25

Recognizing perfect squares on sight is the single biggest speed upgrade for radical problems. These 25 are worth knowing cold:

nn²nn²nn²nn²nn²
11636111211625621441
24749121441728922484
39864131691832423529
416981141961936124576
52510100152252040025625

Square roots of negative numbers (a straight answer about i)

No real number times itself gives a negative: positive × positive is positive, and negative × negative is also positive. So √−16 has no real answer. Rather than shrug, mathematics defines i = √−1 and builds from there: √−16 = √16 × i = 4i. Check it: (4i)² = 16 × i² = 16 × (−1) = −16. These imaginary numbers are not a party trick; they are the backbone of electrical engineering and quantum mechanics. This calculator returns the imaginary result and says so plainly. One nice asymmetry: odd roots do not need i at all. The cube root of −27 is simply −3, because (−3)³ = −27. You will meet i again the moment a quadratic has a negative discriminant, which our quadratic formula calculator also handles honestly.

Estimating a square root by hand

Stranded without a calculator? Sandwich the number between the two nearest perfect squares from the table above. For 72: it sits between 64 (8²) and 81 (9²), so √72 is between 8 and 9. Since 72 is roughly halfway between 64 and 81, guess around 8.5.

Want a decimal place or two more? Use one round of the divide-and-average trick: divide your number by the guess, then average the two. For 72 with guess 8.5: 72 ÷ 8.5 = 8.47, and the average of 8.5 and 8.47 is about 8.485. The true value is 8.48528, so one step of mental arithmetic got you three correct decimals. Each repeat of that step roughly doubles the number of correct digits. If the averaging leaves you with an awkward fraction, our fraction calculator will happily tidy it up.

Frequently asked questions

What is the square root of a number?

The square root of a number is the value that gives you that number when multiplied by itself. The square root of 25 is 5 because 5 x 5 = 25. Every positive number actually has two square roots (5 and -5 both square to 25), but the radical symbol refers to the positive one, called the principal root.

How do I simplify a square root by hand?

Break the number into its prime factors, then pull out one copy of every matched pair. For 72: the factors are 2 x 2 x 2 x 3 x 3. One pair of 2s and one pair of 3s come outside as 2 x 3 = 6, and the leftover 2 stays inside. So the square root of 72 simplifies to 6 times the square root of 2.

What is the square root of a negative number?

No real number squares to a negative, so the answer is imaginary. Mathematicians define i as the square root of -1, and everything follows from there: the square root of -16 is 4i, because (4i) x (4i) = 16 x (-1) = -16. This calculator shows the imaginary result rather than an error.

Is the square root of 2 a rational number?

No. The square root of any whole number that is not a perfect square is irrational: its decimal runs forever without repeating. That is exactly why simplified radical form exists. Writing 6 times the square root of 2 is exact, while 8.48528 is a rounded approximation.

How do I know if a number is a perfect square?

Take its square root: if the result is a whole number, it is a perfect square. There are also quick filters: perfect squares never end in 2, 3, 7, or 8, and in a prime factorization every prime must appear an even number of times. This calculator flags perfect squares and perfect cubes automatically.

What is a cube root and how is it different?

A cube root undoes cubing: the cube root of 27 is 3 because 3 x 3 x 3 = 27. Two differences from square roots matter. Simplifying means pulling out triples of prime factors instead of pairs, and negative numbers are fine: the cube root of -27 is simply -3, no imaginary numbers needed.

How do I estimate a square root without a calculator?

Sandwich it between the two nearest perfect squares. For 72, note that 64 (8 squared) is below it and 81 (9 squared) is above it, so the answer is between 8 and 9, and closer to 8.5 since 72 sits near the middle. One refinement step, averaging 8.5 with 72 divided by 8.5, gets you 8.485, which is accurate to three decimals.

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