Square Root Calculator

Roots, powers, radical simplification, and prime factorization. Results update instantly.

Last reviewed: April 2026
Built and maintained by · Last updated April 2026
Square Root (n=2)
8.485281374
sqrt(72) = 8.485281374...

Roots

Cube Root (n=3)
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Nth Root (n = 2)
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Powers

Squared (n2)
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Cubed (n3)
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Nth Power (n = 2)
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Perfect Square?
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Simplified Radical Form

Simplified sqrt(n)
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Prime Factorization

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Important Disclaimer: The information provided on this page is for educational purposes only. It shouldn't be considered as a substitute for professional advice in mathematics or any other field. Always consult with a qualified professional for accurate and personalized guidance.

About the Square Root Calculator

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 * 4 = 16. Understanding square roots is fundamental in many areas of mathematics, including geometry, algebra, and calculus. They're also used in various real-world applications such as physics, engineering, and finance.

Common Mistakes

One common mistake when dealing with square roots is to think that the square root of a negative number is a real number. However, the square root of a negative number is an imaginary number, specifically a complex number. For example, the square root of -1 is represented as i, where i^2 = -1.

Real-World Applications

1. Engineering and Physics: Square roots are used extensively in physics for calculating speed, velocity, and acceleration. For example, the formula for velocity is v = d/t, where v is velocity, d is distance, and t is time. Taking the square root of both sides can help in solving for time when distance and velocity are known.

2. Finance: In finance, square roots are used in the calculation of volatility and risk. The standard deviation, which measures the dispersion of a set of data points around their mean, is calculated using the square root of the variance. This helps in understanding the risk associated with an investment.

3. Geometry: Square roots are essential in geometry for calculating the length of sides of shapes. For instance, in a right-angled triangle, the Pythagorean theorem (a^2 + b^2 = c^2) uses square roots to find the length of the hypotenuse (c) when the lengths of the other two sides (a and b) are known.

Examples

1. Example 1: Calculate the square root of 81. The square root of 81 is 9 because 9 * 9 = 81.

2. Example 2: Calculate the square root of 12. The square root of 12 is approximately 3.46 because 3.46 * 3.46 ≈ 12.

3. Example 3: Calculate the square root of 0.25. The square root of 0.25 is 0.5 because 0.5 * 0.5 = 0.25.

What Happens If...

1. What happens if you try to find the square root of a negative number? If you try to find the square root of a negative number, you will get an imaginary number. For example, the square root of -9 is 3i because 3i * 3i = -9.

2. What happens if you square a negative number? When you square a negative number, the result is positive. For example, (-3)^2 = 9.

3. What happens if you square a fraction? When you square a fraction, you square both the numerator and the denominator. For example, (1/2)^2 = 1/4.

Frequently Asked Questions

A square root is a value that, when multiplied by itself, gives the original number. A cube root is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 * 3 * 3 = 27.
To find the square root of a number without a calculator, you can use the long division method. This method involves dividing the number into pairs of digits from right to left and then performing a series of steps to find the square root. For example, to find the square root of 256, you would divide 256 into pairs: 2 and 56. Then, you would find the largest number that, when multiplied by itself, gives a number less than or equal to 256. In this case, it's 16 because 16 * 16 = 256.
Yes, a negative number can have a square root, but it's an imaginary number. For example, the square root of -1 is represented as i, where i^2 = -1.
The square root of zero is zero. This is because 0 * 0 = 0.
Square roots are used in various real-life situations, such as calculating the length of sides of shapes, determining the speed of an object, and understanding the risk associated with an investment. For example, in a right-angled triangle, the Pythagorean theorem (a^2 + b^2 = c^2) uses square roots to find the length of the hypotenuse (c) when the lengths of the other two sides (a and b) are known.