Square Root Calculator
Calculate Square Root
About Square Root Calculator
What Is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of 25 is 5 because 5 x 5 = 25. Written mathematically as sqrt(25) = 5 or 25^0.5 = 5. Square roots appear constantly in geometry (diagonal of a square, distance formula), physics (wave equations, standard deviation), engineering (signal processing), and finance (volatility calculations). Perfect squares like 4, 9, 16, 25, and 36 have whole number roots, while most numbers have irrational square roots that continue infinitely without repeating.
Properties of Square Roots
Key properties include: sqrt(a x b) = sqrt(a) x sqrt(b), sqrt(a/b) = sqrt(a)/sqrt(b), sqrt(a^2) = |a|, and (sqrt(a))^2 = a. Negative numbers do not have real square roots since no real number multiplied by itself gives a negative result. The square root of negative numbers involves imaginary numbers (i = sqrt(-1)), which this calculator also handles by displaying results in the form a + bi.
How to Use This Calculator
Enter any number and click Calculate. The tool shows the square root as a decimal (to 10 decimal places), identifies whether the result is rational or irrational, and indicates if the input is a perfect square. For perfect squares, the exact integer root is highlighted. You can also calculate nth roots by specifying the root index.
Frequently Asked Questions
What is a perfect square?
A perfect square is a number whose square root is a whole number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are the squares of integers 1 through 10.
Can you take the square root of a negative number?
Not in the real number system. The square root of a negative number is imaginary. For example, sqrt(-9) = 3i where i is the imaginary unit representing sqrt(-1).
How do I estimate square roots mentally?
Find the two perfect squares your number falls between. For sqrt(50): it is between sqrt(49)=7 and sqrt(64)=8, closer to 7. The actual answer is approximately 7.07.
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