How do I calculate a square root by hand?

Estimate and refine: to find √50, note it lies between 7 and 8, try 7.1, and adjust. Each iteration roughly doubles the accurate digits.

Can you take the square root of a negative number?

Not within real numbers, since no real number squares to a negative. Complex numbers handle it using i, defined as the square root of −1.

Why does √25 give 5 and not −5?

By convention the radical sign denotes the positive root. The equation x² = 25 has both solutions, which is why the quadratic formula includes a plus-or-minus.

What is an irrational root?

One that cannot be written as a fraction, such as √2. Its decimal continues forever without repeating, so any written value is an approximation.

Square Root Calculator - Square, Cube and Nth Roots

Root Calculator

A root asks which number, multiplied by itself a given number of times, produces your value. This calculator handles square, cube and any nth root.

Formula

√a = b where b × b = a · ⁿ√a = a^(1/n)

Every positive number has two square roots, one positive and one negative
The radical sign means the positive one

√50 = √(25 × 2) = 5√2 ≈ 7.0711

Roots Are Fractional Powers

Every root can be written as an exponent:

√a = a^1/2
∛a = a^1/3
The nth root of a = a^1/n

This is why the laws of indices apply to roots as well, and how calculators evaluate them.

Two Answers, One Symbol

Both 5 and −5 square to 25, so the equation x² = 25 has two solutions. But the symbol √25 conventionally denotes only the positive root, 5.

That is why the quadratic formula carries an explicit ±: it has to reintroduce the negative possibility that the radical sign leaves out.

Odd roots behave differently. The cube root of −8 is −2, a perfectly valid real answer, because a negative number cubed stays negative.

Perfect Squares and Irrational Roots

Perfect squares (1, 4, 9, 16, 25, 36...) have whole-number roots. Most numbers do not: √2 = 1.41421356... continues forever without repeating.

Such roots are irrational — they cannot be written as a fraction of two integers. The Pythagoreans reputedly found this discovery deeply unwelcome, since it broke their conviction that all numbers were ratios. Where precision matters, keep √2 in exact form rather than rounding early.

Quick Reference

Roots worth knowing by sight.
n	√n	n²	n³
2	1.4142	4	8
3	1.7321	9	27
5	2.2361	25	125
10	3.1623	100	1,000
16	4.0000	256	4,096

Frequently Asked Questions

How do I calculate a square root by hand?: Estimate and refine: to find √50, note it lies between 7 and 8, try 7.1, and adjust. Each iteration roughly doubles the accurate digits.
Can you take the square root of a negative number?: Not within real numbers, since no real number squares to a negative. Complex numbers handle it using i, defined as the square root of −1.
Why does √25 give 5 and not −5?: By convention the radical sign denotes the positive root. The equation x² = 25 has both solutions, which is why the quadratic formula includes a plus-or-minus.
What is an irrational root?: One that cannot be written as a fraction, such as √2. Its decimal continues forever without repeating, so any written value is an approximation.