Square Root Calculator

Introduction

A square root is a number that, when multiplied by itself, gives you the original number. For example, the square root of 25 is 5, because 5 × 5 = 25. Finding square roots is a key skill in algebra and shows up in many math problems, from solving equations to working with the Pythagorean theorem.

This square root calculator lets you quickly find the square root of any number. Just type in a number, and the tool calculates the result. It handles perfect squares like 16 or 49, as well as numbers that give long decimal answers, like 2 or 7. This calculator saves you time and gives you an accurate answer right away.

How to Use Our Square Root Calculator

Enter any number into the calculator, and it will instantly give you the square root of that number.

Number: Type the number you want to find the square root of. This can be any positive number, including decimals. For example, you could enter 25, 144, or 7.5. The calculator will compute the square root and display the result right away.

What Is a Square Root?

A square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 25 is 5, because 5 × 5 = 25. The symbol for square root is √, and the number inside the symbol is called the radicand. Every positive number actually has two square roots: a positive one and a negative one. So √25 = +5 and −5, since (−5) × (−5) also equals 25. The positive answer is called the principal root, and that is the one most people mean when they say "the square root."

Perfect Squares vs. Non-Perfect Squares

A perfect square is a number whose square root is a whole number. Numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are all perfect squares. When a number is not a perfect square (like 2, 3, 5, or 75), its square root is an irrational number, meaning the decimal goes on forever without repeating. In these cases, you can either leave the answer in simplified radical form or round the decimal. For example, √75 simplifies to 5√3 because 75 = 25 × 3, and √25 = 5. Finding the factors that make simplification possible is closely related to prime factorization, which breaks a number down into its prime building blocks.

How to Simplify a Square Root

To simplify a square root, find the largest perfect square that divides evenly into the radicand. Then break the radicand into two factors: the perfect square and the leftover number. Take the square root of the perfect square and move it outside the radical sign. Here is a quick example:

• √72 → 72 = 36 × 2
• √(36 × 2) = √36 × √2 = 6√2

Cube Roots and Nth Roots

A cube root asks: what number multiplied by itself three times equals the radicand? For instance, the cube root of 125 is 5 because 5 × 5 × 5 = 125. Unlike square roots, cube roots can handle negative numbers and still give a real answer. The cube root of −8 is −2, since (−2) × (−2) × (−2) = −8.

An nth root extends this idea to any root index. The 4th root of 81 is 3 because 3⁴ = 81. The 5th root of 32 is 2 because 2⁵ = 32. When the root index is even and the radicand is negative, there is no real answer. The result involves imaginary numbers, written with the letter i, where i = √(−1).

Key Rules to Remember

• √(a × b) = √a × √b: You can split a radical into the product of two radicals.
• √(a²) = |a|: The square root of a squared number is the absolute value of that number.
• Even roots of positive numbers have two real answers (± values). Odd roots have exactly one real answer.
• A square root can also be written as an exponent: √x = x^1/2, and the nth root of x = x^1/n.

Where Square Roots Are Used

Square roots show up in many areas of math and everyday life. They are used in the Pythagorean theorem to find the length of a side of a right triangle, in the distance formula to measure how far apart two points are, and in quadratic equations when you solve using the quadratic formula. Square roots also appear in statistics. For example, standard deviation involves taking the square root of the variance. Scientists, engineers, and architects all rely on square roots regularly in their work.