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Enter any number, and the calculator will instantly determine its principal square root and raise it to the nth power.
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The Square Root Calculator helps you find the square root and nth root of any number, including negative numbers. It also indicates whether a number is a perfect square. For example, 4, 9, and 16 are perfect squares of 2, 3, and 4, respectively.
Save time and avoid manual errors. You can also check our Exponent Calculator for powers and roots.
The square root of a number x is a number y such that \(y^2 = x\). In other words, it is a factor that, when multiplied by itself, gives the original number.
Example: 3 and -3 are square roots of 9 since \(3^2 = (-3)^2 = 9\).
Formula for nth root:
$$ \sqrt[n]{x} = x^{1/n} $$
The principal square root is the nonnegative square root of a number, denoted as \(\sqrt{x}\). Example: \(\sqrt{49} = 7\). The number under the root is called the radicand.
For perfect squares:
\(\sqrt{25} = \sqrt{5 \times 5} = \sqrt{5^2} = 5\)
For non-perfect squares, approximate using nearby perfect squares. Example: \(\sqrt{54}\)
Example: \(\sqrt{27}\)
\(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\)
For a fraction \(\frac{a}{b}\):
$$ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$
Example: \(\sqrt{\frac{9}{25}} = \frac{3}{5} = 0.6\)
Negative numbers have imaginary roots. Using complex numbers:
$$ i = \sqrt{-1} $$
Examples:
Yes. Positive numbers have two roots: one positive and one negative.
No, it is irrational. It cannot be expressed as a fraction.
Some are rational; others are irrational.
| x | √x |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
| x | √x |
|---|---|
| -4 | 2i |
| -9 | 3i |
| -16 | 4i |
Square roots are fundamental in mathematics, appearing in algebra, physics, and daily life. The online square root calculator provides accurate results quickly for integers, fractions, or negative numbers.
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