Square Root Calculator

Find the square root of any real number, including negative ones, where the answer involves imaginary numbers. Need a cube root or something higher? The radicals and roots calculator handles any index. Going the other direction, the square calculator squares a number instead of rooting it.

Enter a Number

√x =
Enter a negative number to see the complex (imaginary) roots instead.

Example Square Roots

x √x Perfect square?
81±9Yes
25±5Yes
100±10Yes
10±3.16227766No

Result

Principal root:

All roots:

Share this calculation:

Enter a number for x and hit Calculate to see its square root here.

How to Use This Calculator

  1. Type a number into the x field positive, negative, decimal, whatever you need.
  2. Click Calculate. You'll get the principal root plus the second root right below it.
  3. If x is negative, both roots come back as imaginary numbers instead of real ones that's expected, not an error.
  4. Copy the answer or print a clean version using the buttons under the result.

Practical Example / Formula Used

The square root of x is a number a where a × a = x. Every positive number actually has two square roots, one positive and one negative, since both give the same result once squared:

a² = x  →  a = ±√x

For x = 5, that works out to a = ±2.23606798. The positive one, 2.23606798, is called the principal square root — it's what people usually mean when they just say "the square root of 5" without specifying which one.

When x is negative, there's no real number that squares to a negative result (any real number times itself lands at zero or above). So the roots become imaginary instead, written with i where i² = -1. The square root of -5, for instance, comes out to ±2.23606798i.

Why Use This Calculator?

  • It handles negative inputs properly instead of just throwing an error, giving you the actual complex answer.
  • The perfect-square check saves you a mental math step when you're not sure if a number roots out cleanly.
  • Both roots are shown, not just the principal one, which matters if you're solving an equation like x² = 25.
  • Share links let you send someone your exact calculation with the numbers already filled in.

Key Factors / Variables Explained

x - the number you're finding the square root of, sometimes called the radicand. It can be any real number.

Principal root - the non-negative square root, and the one meant by default when someone writes √x without a ± sign in front.

Perfect square - a number whose square root is a whole number. 81, 25, and 100 are perfect squares because their roots (9, 5, and 10) are all integers.

Imaginary/complex root - what you get when x is negative. Written with i, where i represents √-1.

Frequently Asked Questions

Because squaring removes the sign. Both 3 × 3 and -3 × -3 equal 9, so 9 has two numbers that square to it: 3 and -3. The positive one is called the principal root and is what most calculators return by default.

You'll get an imaginary result, written with i. There's no real number that squares to a negative value, so mathematicians use i (where i² = -1) to represent those answers instead of saying there's no solution at all.

Memorizing the squares of 1 through 15 or so covers most everyday cases. Past that, estimate between two known squares — if a number falls between 100 (10²) and 121 (11²) and isn't a whole-number match, it's not a perfect square.

Yes, and it's just 0. Since 0 × 0 = 0, there's only one root here instead of the usual positive/negative pair.

A square root is just an nth root where n is fixed at 2. If you need a cube root, 5th root, or anything with a different index, our radicals and roots calculator covers those.

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