Square root principle calculator

Square root principle calculator DEFAULT

Square Root Calculator

Calculator Use

Use this calculator to find the principal square root and roots of real numbers. Inputs for the radicand x can be positive or negative real numbers. The answer will also tell you if you entered a perfect square.

The answer will show you the complex or imaginary solutions for square roots of negative real numbers.  See also the Simplify Radical Expressions Calculator to simplify radicals instead of finding fractional (decimal) answers.

Square Roots, odd and even:

There are 2 possible roots for any positive real number. A positive root and a negative root. Given a number x, the square root of x is a number a such that a2 = x. Square roots is a specialized form of our common roots calculator.

"Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3, since (-3)2 = (+3)2 = 9. Any nonnegative real number x has a unique nonnegative square root r; this is called the principal square root For example, the principal square root of 9 is sqrt(9) = +3, while the other square root of 9 is -sqrt(9) = In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root."[1].

Perfect Square Calculator

This calculator will also tell you if the number you entered is a perfect square or is not a perfect square.  A perfect square is a number x where the square root of x is a number a such that a2 = x and a is an integer. For example, 4, 9 and 16 are perfect squares since their square roots, 2, 3 and 4, respectively, are integers.

Example Square Roots:

  • The 2nd root of 81, or 81 radical 2, or the square root of 81 is written as $$ \sqrt[2]{81} = \sqrt[]{81} = \pm 9 $$.
  • The 2nd root of 25, or 25 radical 2, or the square root of 25 is written as $$ \sqrt[2]{25} = \sqrt[]{25} = \pm 5 $$.
  • The 2nd root of , or radical 2, or the square root of is written as $$ \sqrt[2]{} = \sqrt[]{} = \pm 10 $$.
  • The 2nd root of 10, or 10 radical 2, or the square root of 10 is written as $$ \sqrt[2]{10} = \sqrt[]{10} = \pm $$.

To calculate fractional exponents use our calculator for Fractional Exponents.


[1] Weisstein, Eric W. "Square Root." From MathWorld -- A Wolfram Web Resource. Square Root

Additional reading on square roots:

At Math is Fun: square root


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\mathrm{lcm} \mathrm{gcf} \mathrm{factors} \mathrm{long\:addition} \mathrm{scientific}
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Calculator Tutorial 9: Square roots on a scientific calculator

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