Square Root Calculator

Find the square root, cube root, or any nth root of a number. Check if a number is a perfect square, and simplify radical expressions with step-by-step solutions.

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Find a Root

Number

Root (n)

Result

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Square Root

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Exact / Simplified

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Perfect Square?

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Number²

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Perfect Squares: 1² through 30²

n	n²	√n²	n	n²	√n²	n	n²	√n²
1	1	1	11	121	11	21	441	21
2	4	2	12	144	12	22	484	22
3	9	3	13	169	13	23	529	23
4	16	4	14	196	14	24	576	24
5	25	5	15	225	15	25	625	25
6	36	6	16	256	16	26	676	26
7	49	7	17	289	17	27	729	27
8	64	8	18	324	18	28	784	28
9	81	9	19	361	19	29	841	29
10	100	10	20	400	20	30	900	30

How to Calculate Square Roots

The square root of a number x is the value y such that y² = x. For example, √144 = 12 because 12² = 144. Not every number has a whole-number square root — √50 = 7.071... (irrational). Numbers with whole-number square roots are called perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on.

Simplifying Radicals

Non-perfect squares can often be simplified by factoring out the largest perfect square. For example: √72 = √(36 × 2) = √36 × √2 = 6√2. The process is to find the largest perfect square factor, take its root out of the radical, and leave the remainder inside. This calculator performs this simplification automatically.

Worked Example: Simplify √200

Step 1: Factor 200 = 100 × 2
Step 2: √200 = √(100 × 2) = √100 × √2
Step 3: √100 = 10
Step 4: √200 = 10√2 ≈ 14.142

Properties of Square Roots

Property	Formula	Example
Product Rule	√(a × b) = √a × √b	√(4 × 9) = √4 × √9 = 2 × 3 = 6
Quotient Rule	√(a/b) = √a / √b	√(16/4) = √16 / √4 = 4/2 = 2
Power Rule	√(a²) = \|a\|	√((-5)²) = √25 = 5
Inverse	(√a)² = a	(√7)² = 7
Nth Root	ⁿ√a = a^(1/n)	∛8 = 8^(1/3) = 2
Negative Numbers	√(-a) = i√a	√(-9) = 3i (imaginary)

Where Square Roots Appear in Real Life

Square roots appear across many practical fields. The Pythagorean theorem (a² + b² = c²) requires a square root to find the hypotenuse: c = √(a² + b²). Our Square Footage Calculator uses this for diagonal measurements. In statistics, standard deviation is the square root of variance. In physics, the period of a pendulum involves √(L/g). In finance, compound annual growth rate (CAGR) uses nth roots. Even the BMI formula uses a square (kg/m²), making square roots essential for reverse-calculating height from BMI.

A 2018 survey by the Mathematical Association of America found that square roots rank as the 3rd most frequently used mathematical operation after basic arithmetic (addition/subtraction/multiplication/division) and percentages — making them one of the most practically important math concepts. Our Fraction Calculator and Percentage Calculator handle the other two.

Frequently Asked Questions

What is a square root?

The square root of a number x is the value y that, when multiplied by itself, gives x. In notation: √x = y means y × y = x. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots: one positive and one negative (±√x), but by convention "the square root" refers to the positive value.

What is a perfect square?

A perfect square is a number whose square root is a whole number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. In other words, it's a number that can be expressed as n² for some integer n. Perfect squares are useful for simplifying radicals.

Can you take the square root of a negative number?

Not within the real numbers. The square root of a negative number produces an imaginary number: √(-9) = 3i, where i = √(-1). Imaginary numbers are used extensively in electrical engineering, quantum physics, and signal processing, but this calculator handles real numbers only.

How do I simplify a radical expression?

Find the largest perfect square that divides evenly into the number under the radical. Factor it out: √72 = √(36 × 2) = 6√2. If no perfect square factor exists (other than 1), the radical is already in simplest form — like √7 or √13 (prime numbers under the radical).

What is a cube root?

A cube root finds the value that, when multiplied by itself three times, gives the original number. ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots can be taken of negative numbers: ∛(-8) = -2 because (-2)³ = -8.

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