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Square Root Calculator

Square roots appear in geometry, physics, engineering, and everyday problem-solving. Given a number, you need its square root—the value that, when multiplied by itself, returns the original. This calculator instantly computes both the principal (positive) and negative square roots, handles decimals and large numbers, and shows whether a given number is a perfect square. Ideal for students, engineers, and professionals who need rapid, accurate results without manual calculation.

Last updated: May 10, 2026

Creators Jasmine J. Mah, BSc

Reviewers Mateusz Mucha and Anna Szczepanek

752 people find this calculator helpful

Understanding Square Roots

A square root is one of the inverse operations in algebra. If y² = x, then y is a square root of x. The notation √x denotes the principal (non-negative) square root.

Every positive real number actually possesses two square roots: one positive and one negative. For example, both 5 and −5, when squared, equal 25. However, by convention, √25 refers only to the positive root, 5.

Perfect squares are whole numbers whose square roots are also whole numbers:

√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5

Most numbers are not perfect squares. Their roots are irrational—they cannot be expressed as simple fractions and their decimal expansions never terminate. For instance, √2 ≈ 1.41421 (a value known to Babylonian mathematicians around 1800 BCE).

The Square Root Formula

The square root relationship is defined as:

If y = √x, then y² = x

Equivalently: y = ±√x ⟺ y² = x

x — The number under the square root (the radicand)
y — The square root value; when y is squared, it equals x

Finding and Simplifying Square Roots

When you lack a calculator, estimation works well for many purposes. Find the two nearest perfect squares that bracket your target number. The square root will lie between their roots.

For example, since 26 sits between 25 and 36, we know √26 falls between 5 and 6. Since 26 is much closer to 25, √26 ≈ 5.1.

To refine an estimate using the Babylonian method:

Start with an initial guess.
Divide your target number by the guess.
Average the guess and the quotient.
Repeat until precision plateaus.

Simplifying radicals means rewriting √x as a√b where b is as small as possible. For instance, √8 = √(4 × 2) = 2√2. This form is 'simpler' because the radicand is smaller.

Operations with Square Roots

Square roots follow specific algebraic rules that differ from simple arithmetic:

Multiplication: √a × √b = √(a × b). For example, √2 × √8 = √16 = 4.

Division: √a ÷ √b = √(a ÷ b). So √20 ÷ √5 = √4 = 2.

Addition and subtraction: These operations require like radicals—identical radicands. You cannot simplify √2 + √3 further. However, 3√5 + 2√5 = 5√5 because the radicands match.

Fractional exponents: √x can be written as x^1/2. This notation proves useful in calculus and higher algebra, where the power rule applies: √(xⁿ) = x^n/2.

Key Pitfalls and Practical Notes

Avoid common mistakes when working with square roots:

Square root of negative numbers — In real numbers, the square root of a negative number is undefined. Mathematicians introduced complex numbers to extend beyond this limit, where √(−1) = i. Unless your problem explicitly involves complex numbers, assume all radicands are non-negative.
Confusing √4 with ±2 — The symbol √ always means the principal (positive) root. √4 = 2, not ±2. If an equation asks for all square roots of 4, the answer is ±2. But √4 alone is always 2.
Rationality of roots — Not all square roots simplify to whole numbers or fractions. √2, √3, √5, and √7 are irrational. If your answer must be rational, verify whether the radicand contains only even prime powers (e.g., 4 = 2², 36 = 6²).
Square roots of decimals — To find √0.36, convert to a fraction first: 0.36 = 36/100. Then √(36/100) = 6/10 = 0.6. This method avoids confusion with decimal approximations.

Frequently Asked Questions

Do positive numbers have more than one square root?

Yes. Every positive real number has exactly two square roots: one positive and one negative. For instance, 36 has roots 6 and −6. When you square either value, you recover 36 because (−6)² = 36, just as 6² = 36. By convention, √36 refers only to the principal root, 6. To denote both roots explicitly, write ±√36.

How can I calculate a square root by hand?

The Babylonian method, also called Newton's method, is efficient. Make an initial guess, then repeatedly divide your target by the guess and average the result with the guess. For √10 starting with a guess of 3: (10÷3 + 3)÷2 ≈ 3.17. Repeating yields 3.162..., which is accurate to three decimal places. Each iteration roughly doubles your precision. This works for any positive number.

What makes some square roots rational and others irrational?

A square root is rational if and only if the radicand is a perfect square. For example, √9 = 3 and √(4/9) = 2/3 are both rational. Conversely, √2, √3, and √5 are irrational because 2, 3, and 5 cannot be expressed as ratios of perfect powers. Irrational roots never yield repeating or terminating decimals.

How do I eliminate square roots from an equation?

Square both sides of the equation. If √x = 5, squaring gives x = 25. This removes the radical sign. However, squaring can introduce extraneous solutions, so always verify your answer in the original equation. For example, if √(2x) = 6, squaring gives 2x = 36, so x = 18. Checking: √(2×18) = √36 = 6. ✓

Can I add or subtract square roots with different radicands?

Not directly. √2 + √3 cannot be simplified further because the radicands differ. You can only combine like radicals—terms with identical radicands. For instance, 2√5 + 3√5 = 5√5. If you need a decimal answer, compute √2 ≈ 1.414 and √3 ≈ 1.732, then add: 1.414 + 1.732 ≈ 3.146.

What is the derivative of a square root function?

The derivative of f(x) = √x is f'(x) = 1/(2√x), or equivalently x⁻¹/². This comes from rewriting √x as x^(1/2) and applying the power rule: d/dx[x^n] = nx^(n−1). Thus d/dx[x^(1/2)] = (1/2)x^(−1/2) = 1/(2√x). This derivative approaches infinity as x approaches 0 from the right, reflecting the steep curve near the origin.

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