Square Root of a Number: How to Find It

This calculator finds the square root of any non-negative number you enter. Type in a value, and you get its principal square root instantly, along with the option to see the result rounded to a chosen number of decimal places. The square root of a number x is the value that, when multiplied by itself, equals x; for example, the square root of 16 is 4 because 4 × 4 = 16.

Formula

The square root is written with a radical symbol: √x. The formula is:

√x = y where y² = x and y ≥ 0.

Here, x is the number you want the root of (the radicand), and y is the result. The principal square root is always non-negative. For example, both 5 and −5 squared equal 25, but the principal square root is 5, which is what calculators and this tool return.

Worked Example

Using the default value of 25:

1. Start with x = 25.
2. Find a number that, when multiplied by itself, equals 25.
3. 5 × 5 = 25, so √25 = 5.

The calculator returns 5, matching the manual calculation.

When to Use the Square Root

Square roots appear in geometry (finding side lengths of squares from area), physics (calculating velocity from kinetic energy), statistics (standard deviation is the square root of variance), and finance (volatility measures). They are also fundamental in solving quadratic equations via the quadratic formula, where the discriminant involves a square root.

You would use this calculator whenever you need the square root of a number quickly, whether for homework, work, or personal projects. It handles perfect squares (like 144) and non-perfect squares (like 2) equally well, returning a decimal approximation for the latter.

How to Find Square Roots Manually

There are several methods to compute a square root without a calculator:

• Guess and check: Estimate, square it, adjust, repeat. For √20, try 4.5 (20.25), then 4.4 (19.36), so the root is about 4.47.
• Babylonian method (Newton’s method): Start with a guess g, then replace g with (g + x/g)/2. Repeat until the change is tiny. For √25 with guess 3: (3 + 25/3)/2 = 5.67, then (5.67 + 25/5.67)/2 ≈ 5.02, then ≈ 5.00004.
• Prime factorization: Break the number into prime factors and pair them. For 144 = 2â�´ × 3², take one from each pair: 2² × 3 = 12.

These methods are useful when you don’t have a calculator, but for most purposes, using the tool is faster and less error-prone.

Common Mistakes and Misconceptions

One frequent mistake is forgetting that the square root of a positive number has two roots: one positive and one negative. However, the symbol √x by definition means the non-negative root. So √9 = 3, not ±3. If you need both, you write ±√9.

Another error is mixing up the square root with the cube root or higher roots. The square root specifically asks for the number that squares to the original. Also, be careful with units: if you take the square root of an area in square meters, the result is in meters. For example, √(25 m²) = 5 m.

Rounding too early can also cause inaccuracies. If you round √2 to 1.4 and then square it, you get 1.96, not 2. Only round at the final step.

Limits of the Formula

The square root of a negative number is not a real number; it is an imaginary number. For example, √(−4) = 2i, where i is the imaginary unit. This calculator does not handle imaginary numbers—it only accepts non-negative inputs. If you need to work with negative radicands, you must use complex numbers.

Also, the square root of a non-perfect square is irrational—it cannot be expressed as a simple fraction and has an infinite, non-repeating decimal expansion. The calculator gives a decimal approximation, which is sufficient for most practical purposes.

Frequently Asked Questions

What is the square root of 0?

The square root of 0 is 0, because 0 × 0 = 0. This is a special case where the number is its own square root.

Can you take the square root of a negative number?

In real numbers, no. The square root of a negative number is not defined. In complex numbers, it is defined using the imaginary unit i, where i² = −1. For example, √(−9) = 3i.

What is the difference between square root and squared?

Squaring a number means multiplying it by itself (e.g., 4² = 16). Taking the square root is the inverse operation: finding the number that, when squared, gives the original. So √16 = 4.

How do you simplify a square root like √72?

To simplify, factor out perfect squares. √72 = √(36 × 2) = √36 × √2 = 6√2. This is the exact simplified form; the decimal approximation is about 8.485.

What is the square root of a fraction?

You can take the square root of the numerator and the denominator separately. For example, √(4/9) = √4 / √9 = 2/3. This works because the square root of a quotient is the quotient of the square roots.

Why is the square root of 2 irrational?

√2 cannot be written as a fraction of two integers. A classic proof assumes it can be written as a/b in lowest terms, then shows both a and b must be even, contradicting the assumption. This means its decimal expansion goes on forever without repeating.

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in x | Number | - | 25 | 0
out root | Square root | - | sqrt(x) | 4