Square Root Calculator

What Is a Square Root?

A square root of a number is a value that, when multiplied by itself, produces the original number. The square root of 25 is 5 because 5 times 5 equals 25. The square root of 64 is 8 because 8 times 8 equals 64. The square root of 100 is 10 because 10 times 10 equals 100. The mathematical symbol for square root is the radical sign (√), and the operation is the inverse of squaring a number. Enter any number in the calculator above to find its square root instantly. You can also use the URL pattern ?n=NUMBER for a shareable calculation, for example square root of 5000 or square root of 3.75.

How to Find the Square Root of a Number?

There are several methods to find the square root of a number. The simplest is to use the find square root calculator above. For manual calculation, you can use the prime factorization method: break the number into its prime factors, pair them up, and take one from each pair. For example, the square root of 36 equals the square root of (2 times 2 times 3 times 3), which gives you 2 times 3 equals 6. For numbers that are not perfect squares, estimation and long division methods work but are tedious.

A calculator gives you the decimal value instantly. The square root of 2 is approximately 1.4142, the square root of 3 is approximately 1.7321, and the square root of 5 is approximately 2.2361. The root value of any positive number is unique and well-defined.

Calculation for Square Root of Popular Numbers

Below are the most frequently calculated square roots, sorted from smallest to largest. Click any row to load that value into the square root calculator above and see the full-precision result.

Square Root of Decimal Numbers

The same algorithm works for decimal numbers, including very small fractions and irrational inputs. Numbers are sorted from smallest to largest. Click any row to recalculate with full precision.

Common Square Roots Reference Table

Here are the square roots of the most frequently searched numbers. Perfect squares produce whole number results while other numbers produce irrational decimals that continue infinitely without repeating.

Perfect squares: The square root of 1 is 1. The square root of 4 is 2. The square root of 9 is 3. The square root of 16 is 4. The square root of 25 is 5. The square root of 36 is 6. The square root of 49 is 7. The square root of 64 is 8. The square root of 81 is 9. The square root of 100 is 10. The square root of 121 is 11. The square root of 144 is 12. The square root of 169 is 13. The square root of 196 is 14. The square root of 225 is 15. The square root of 256 is 16. The square root of 289 is 17. The square root of 324 is 18. The square root of 361 is 19. The square root of 400 is 20.

Non-perfect squares (to 4 decimal places): The square root of 2 is 1.4142. The square root of 3 is 1.7321. The square root of 5 is 2.2361. The square root of 6 is 2.4495. The square root of 7 is 2.6458. The square root of 8 is 2.8284. The square root of 10 is 3.1623. The square root of 12 is 3.4641. The square root of 15 is 3.8730. The square root of 17 is 4.1231. The square root of 18 is 4.2426. The square root of 20 is 4.4721. The square root of 27 is 5.1962. The square root of 32 is 5.6569. The square root of 45 is 6.7082. The square root of 48 is 6.9282. The square root of 50 is 7.0711. The square root of 72 is 8.4853. The square root of 75 is 8.6603. The square root of 117 is 10.8167. The square root of 175 is 13.2288.

What Are Perfect Squares?

A perfect square is a number whose square root is a whole number. The first twenty perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, and 400. Recognizing perfect squares helps with mental math and simplifying expressions. If a number is a perfect square, its square root is clean and exact. If it is not a perfect square, its square root is an irrational number, a decimal that never ends and never repeats. Knowing which numbers are perfect squares is fundamental for algebra, geometry, and standardized test preparation.

Simplify Square Root Calculator

To simplify a square root, find the largest perfect square factor of the number inside the radical. The square root of 48 can be simplified because 48 equals 16 times 3. Since the square root of 16 is 4, the square root of 48 simplifies to 4√3. Similarly, the square root of 32 simplifies to 4√2 (because 32 equals 16 times 2). The square root of 45 simplifies to 3√5 (because 45 equals 9 times 5). The square root of 50 simplifies to 5√2 (because 50 equals 25 times 2).

The simplified square root calculator approach used here works on any radical:

• Find the largest perfect square that divides the number under the radical
• Take its square root out of the radical sign
• Leave the remaining factor inside the radical
• Combine into the form a√b where b has no remaining perfect-square factors

This technique is essential in algebra when working with radical expressions. The decimal output from the calculator above and the simplified radical form represent the same value, just in different notations.

Square Root Fractions Calculator

The square root of a fraction is the square root of the numerator divided by the square root of the denominator. For example, the square root of 9/16 equals √9 / √16 = 3/4. The square root of 36/49 equals √36 / √49 = 6/7. For irrational results, simplify the radical first, then rationalize the denominator if needed. The square root of 2/9 equals √2 / 3. The square root of 5/16 equals √5 / 4. To compute the square root of any fraction in the calculator above, divide first and enter the decimal value, then read the answer.

Adding and Subtracting Square Roots

Square roots add and subtract only when they have the same radicand (the number under the radical sign). For example, 2√3 + 5√3 = 7√3, just like 2x + 5x = 7x. But √2 + √3 cannot be combined into a single radical. To use the add square roots calculator approach manually:

• Simplify each radical to its lowest form
• If the simplified radicals match, add or subtract the coefficients
• If they do not match, the expression cannot be combined further

Example: √48 + √27 = 4√3 + 3√3 = 7√3 (since √48 = 4√3 and √27 = 3√3). To get a numeric answer, compute each square root in the calculator above and add the decimals. The three square root calculator question (combining three radicals) follows the same rule: match radicands, then add coefficients.

Square Root of Negative Numbers

Negative numbers do not have real square roots because no real number multiplied by itself produces a negative result. The square root of a negative number involves imaginary numbers: the square root of -1 is defined as i (the imaginary unit). So the square root of -25 equals 5i, and the square root of -49 equals 7i. The general formula is √(-n) = i√n for any positive n. Complex numbers extend this idea further, but for most algebra and geometry work, an imaginary result is the answer. This is the same logic used inside any square root of negative number calculator.

Properties of Square Roots

Square roots follow several mathematical properties that are useful in calculation and algebra. The square root of a product equals the product of the square roots: √(a × b) = √a × √b. The square root of a fraction equals the square root of the numerator divided by the square root of the denominator. Squaring a square root returns the original number (square root squared calculator: (√n)² = n). The square root of zero is zero.

Square Root Calculator Step by Step

For learners who want a square root calculator step by step, here is the long division method, the classic manual technique:

1. Pair the digits of the number from the decimal point outward. For 12345, group as 1 | 23 | 45.
2. Find the largest integer whose square is less than or equal to the leftmost group. Write it as the first digit of the answer.
3. Subtract its square from the leftmost group, then bring down the next pair.
4. Double the current answer, then find a digit X such that (current_answer × 10 + X) × X is less than or equal to the new working number.
5. Append X to the answer and repeat for the next pair.

The Babylonian method (also called Heron's method) is faster: start with a guess g, compute (g + n/g)/2 as the next guess, and repeat. Each iteration roughly doubles the precision. For example, to find √50 starting with g = 7: next = (7 + 50/7)/2 = 7.07142..., then (7.07142 + 50/7.07142)/2 = 7.07107, which matches √50 to 5 decimal places after two iterations. The calculator above uses the IEEE 754 floating-point sqrt operation, which is faster and more accurate than any manual method. For learners who want a full square root calculator with steps shown, work through the long-division procedure above on paper and verify each digit against the calculator result.

Square Roots in Real Life

Square roots appear throughout daily life and professional fields. In construction, the Pythagorean Theorem uses square roots to calculate diagonal measurements and verify right angles. In finance, the standard deviation (a measure of investment risk) involves square roots. In physics, calculating distances, velocities, and wave frequencies requires square root operations. Screen sizes are measured diagonally using the square root of the sum of squared width and height. GPS navigation calculates distances between coordinates using square roots. Even cooking involves square roots when scaling recipes for different pan sizes, since pan area scales with the square of the diameter.

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