Square Root Calculator
Need to find the square root of a number? Enter any value and get the result instantly. This calculator also handles nth roots and verifies your answer by squaring it back.
Understanding Square Roots
The square root operation is the inverse of squaring a number. When you square a number, you multiply it by itself. When you take a square root, you find what number was multiplied by itself to produce the original value.
Every positive number has two square roots: one positive and one negative. For instance, both 5 and -5 are square roots of 25, because 5² = 25 and (-5)² = 25. By convention, the √ symbol refers to the principal (positive) square root. This calculator returns the positive root.
The mathematical notation √x reads as "the square root of x" or "root x". The horizontal bar extends over the entire expression under the radical. For more complex expressions like √(16 + 9), you must compute what's inside the radical first, then take the square root of the result.
Calculating Nth Roots
While square roots are the most common, you can extract any root from a number. The cube root ³√ finds what number cubed equals your input. The fourth root ⁴√ finds what number raised to the fourth power equals your input, and so on.
The general formula for nth roots is ⁿ√x = x^(1/n). This means taking the nth root is the same as raising the number to the power of 1/n. For example, the cube root of 8 is 8^(1/3) = 2, because 2³ = 8.
Higher-order roots produce smaller results. The square root of 16 is 4, but the fourth root of 16 is only 2. As the root index increases, the result approaches 1 for any number greater than 1. This relationship is useful in exponential decay calculations and compound interest problems.
Practical Applications of Square Roots
Square roots appear throughout mathematics, science, and engineering. The Pythagorean theorem uses square roots to find the hypotenuse of right triangles. Standard deviation in statistics requires taking the square root of variance. Electrical engineers use square roots when calculating RMS voltage and current.
In physics, the period of a pendulum involves the square root of its length. The speed of sound in a gas depends on the square root of temperature. Gravitational escape velocity formulas include square roots of planetary radius and mass.
Financial calculations use square roots too. The volatility of investments is often expressed as the square root of variance. The Black-Scholes option pricing model includes square roots of time to expiration. Even simple percentage calculations sometimes require extracting roots to find average growth rates over multiple periods.
Frequently Asked Questions
What is a square root?
A square root of a number x is a value that, when multiplied by itself, equals x. For example, the square root of 16 is 4 because 4 × 4 = 16.
Can you take the square root of a negative number?
In real numbers, no. The square root of a negative number is imaginary. This calculator works with real numbers only, so negative inputs will return zero or an error.
What is the difference between √ and ⁿ√?
The square root symbol √ specifically means the 2nd root. The nth root ⁿ√ lets you specify any index: cube root (n=3), fourth root (n=4), etc.
How accurate are the results?
Results are rounded to four decimal places (0.0001 precision), which is sufficient for most practical applications including engineering and scientific calculations.
What is the square root of zero?
The square root of zero is zero. This is because 0 × 0 = 0. It's the only number that equals its own square root.