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Square Root Calculator - Calculate Roots and Simplify Radicals Online

Free online square root calculator. Calculate square roots, cube roots, nth roots, and simplify radicals instantly. Get step-by-step solutions with simplified radical form.

8 min read

iAbout This Calculator

The square root calculator is an essential mathematical tool for students, engineers, and professionals. Whether you need to find the square root of a number, calculate cube roots, or simplify complex radicals, our free online calculator provides instant, accurate results. Understanding square roots is fundamental to algebra, geometry, physics, and many real-world applications. This calculator not only gives you the decimal answer but also shows the simplified radical form when possible, helping you understand the mathematical structure behind the numbers.

?How to Use

  1. 1

    Select the calculation mode: Square Root for basic square roots, or Nth Root for cube roots, fourth roots, and beyond.

  2. 2

    Enter the number you want to find the root of in the Number field. The calculator accepts positive numbers, zero, and negative numbers (for odd roots).

  3. 3

    For Nth Root mode, enter the root index (n). Use 3 for cube root, 4 for fourth root, etc.

  4. 4

    Click Calculate to see the result instantly. The calculator shows both the decimal approximation and simplified radical form.

  5. 5

    Review the step-by-step breakdown to understand how the calculation was performed and how the radical was simplified.

fFormula

\sqrt[n]{x} = x^{\frac{1}{n}}

The nth root of a number x is equivalent to raising x to the power of 1/n. For square roots (n=2), this means x^(1/2). For cube roots (n=3), this means x^(1/3). The calculator also simplifies radicals by factoring out perfect powers.

x
The number under the radical (radicand)
n
The root index (2 for square root, 3 for cube root, etc.)
result
The number that, when raised to the nth power, equals x

Examples

Perfect Square Root

Inputs: number: 144
12

144 is a perfect square because 12 x 12 = 144. The square root of 144 is exactly 12.

Simplifying a Radical

Inputs: number: 72
6 x square root of 2

72 = 36 x 2, and 36 is a perfect square. So square root of 72 = square root of (36 x 2) = 6 x square root of 2, approximately 8.485.

Cube Root

Inputs: number: 27, nthRoot: 3
3

The cube root of 27 is 3 because 3 x 3 x 3 = 27.

Non-Perfect Root

Inputs: number: 7
2.6457...

7 is not a perfect square and cannot be simplified. The square root is approximately 2.6457513...

Use Cases

Geometry and Distance

Calculate diagonal lengths, distances between points using the Pythagorean theorem, and areas of shapes. Square roots are essential for converting between area and side length.

Physics and Engineering

Compute wave frequencies, electrical impedance, standard deviations, and root mean square values. Many physical formulas involve square roots.

Finance and Statistics

Calculate volatility, standard deviation, and Sharpe ratios. Financial analysis frequently requires square root calculations for risk assessment.

Academic Studies

Solve quadratic equations, simplify algebraic expressions, and work with irrational numbers. Understanding radicals is crucial for higher mathematics.

Construction and Design

Determine dimensions from area specifications, calculate roof pitches, and verify right angles using the 3-4-5 rule and its variants.

Frequently Asked Questions

What is a square root?
A square root of a number x is a value that, when multiplied by itself, gives x. For example, the square root of 25 is 5 because 5 x 5 = 25. Every positive number has two square roots (positive and negative), but by convention, 'the square root' refers to the positive one.
How do you simplify a square root?
To simplify a square root, factor the number under the radical into a product where one factor is a perfect square. Extract the square root of the perfect square and leave the rest under the radical. For example: square root of 72 = square root of (36 x 2) = 6 x square root of 2.
Can you take the square root of a negative number?
In the real number system, you cannot take the square root of a negative number because no real number multiplied by itself gives a negative result. However, in the complex number system, the square root of -1 is defined as i (the imaginary unit), and square root of -x = i x square root of x.
What is the difference between a square root and a cube root?
A square root finds what number multiplied by itself twice equals the radicand (x^2 = radicand). A cube root finds what number multiplied by itself three times equals the radicand (x^3 = radicand). Cube roots can be taken of negative numbers because a negative times a negative times a negative is negative.
What is a perfect square?
A perfect square is an integer that is the square of another integer. Examples include 1 (1x1), 4 (2x2), 9 (3x3), 16 (4x4), 25 (5x5), etc. The square root of a perfect square is always a whole number.

Conclusion

Understanding and calculating square roots is fundamental to mathematics and its applications in science, engineering, and everyday life. Our square root calculator makes it easy to find roots, simplify radicals, and understand the step-by-step process. Whether you are solving homework problems, designing a project, or analyzing data, bookmark this tool for quick access to accurate root calculations.