Square Root Calculator

How to simplify a square root

A simplified radical is the exact answer. √72 as a decimal is 8.4853, which is rounded and so slightly off, while 6√2 is exactly right. The method is the same every time:

1. Break the number under the root into prime factors.
2. Circle each pair of equal factors. Every pair comes out of the root as one factor.
3. Multiply the factors that came out. Multiply the leftovers that stayed in.

Take √72. Its prime factors are 2 × 2 × 2 × 3 × 3. There is one pair of 2s and one pair of 3s, so 2 × 3 = 6 comes out. One 2 has no partner and stays in. √72 = 6√2. For √50, the factors are 2 × 5 × 5, the pair of 5s comes out and √50 = 5√2. For √75, the factors are 3 × 5 × 5, so √75 = 5√3.

If no factor appears twice, the number is already in simplest form. √30 = √(2 × 3 × 5) cannot be simplified. The table has the roots people look up most.

Two square roots multiply the easy way: √a × √b = √(a × b), so √2 × √8 = √16 = 4. Roots add only when the part under the root matches: 3√2 + 5√2 = 8√2, and √8 + √18 = 2√2 + 3√2 = 5√2 once each one is simplified. √2 + √3 cannot be combined.

Perfect squares from 1 to 20

Learn the first dozen and you will spot most clean roots on sight. A square always ends in 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8 is never a perfect square. Ending in the right digit does not make it one, since 24 and 26 are not squares either.

Worked example: √2 with Newton's method

Estimate √2 without a calculator, using xₙ₊₁ = 0.5 × (xₙ + 2/xₙ), starting from a rough guess.

Starting guess: x₀ = 1.5 (a reasonable estimate, since 1.5² = 2.25, close to 2).

Iteration 1: x₁ = 0.5 × (1.5 + 2/1.5) = 0.5 × (1.5 + 1.3333) = 1.41667.
Iteration 2: x₂ = 0.5 × (1.41667 + 2/1.41667) = 1.414216.
Iteration 3: x₃ = 0.5 × (1.414216 + 2/1.414216) = 1.4142136.

After three iterations the estimate matches the true value of √2 ≈ 1.41421356 to seven decimal places. Newton's method roughly doubles the number of correct digits with each iteration, which is why it is the approach most calculators and computers use internally.

Long division method for a perfect square

Before calculators, students found roots digit by digit. For √1,849: split the digits into pairs from the right, 18 and 49. The largest square not above 18 is 16, so the first digit is 4. Subtract 16 from 18 to leave 2, bring down 49 to make 249, and double the 4 to get 8_. Find the digit d where (80 + d) × d fits into 249. That is d = 3, since 83 × 3 = 249 exactly. √1,849 = 43. The same steps carry on past the decimal point for non-perfect squares: bring down pairs of zeros.

Square roots of negative numbers, fractions and decimals

Negative numbers

No real number squared is negative, so √−9 has no real answer. Mathematicians define i = √−1, which gives √−9 = 3i and √−72 = 6√2 i. These are imaginary numbers. The calculator writes a negative input this way and labels the result. Cube roots do not have this problem: ∛(−27) = −3, since (−3) × (−3) × (−3) = −27.

Fractions

Take the root of the top and bottom separately. √(9/16) = 3/4 and √(2/9) = √2/3. If the bottom is not a perfect square, multiply the top and bottom by the root to clear it: 1/√2 = √2/2.

Decimals

Type the decimal straight in. To do one by hand, turn it into a fraction over a power of 10: √0.25 = √(25/100) = 5/10 = 0.5, and √0.0049 = √(49/10,000) = 7/100 = 0.07. The root of a number between 0 and 1 is bigger than the number itself: √0.25 = 0.5, not 0.05.

Cube roots and other nth roots

The cube root ∛n is the number that gives n when multiplied by itself three times: ∛64 = 4, since 4 × 4 × 4 = 64. The nth root generalizes this. The fourth root of 81 is 3, because 3⁴ = 81, and the fifth root of 32 is 2, because 2⁵ = 32. Type the index into the Root Index box above to get any of these. An even root of a negative number is imaginary. An odd root of a negative number is real and negative.

Where square roots show up in everyday calculations

How is the Pythagorean theorem a square-root problem?

The diagonal of a rectangle (or the hypotenuse of a right triangle) with legs of 6 and 8 comes from c = √(a² + b²) = √(36 + 64) = √100 = 10. That is a clean example. Most real measurements give a non-perfect square, such as legs of 5 and 7, where c = √74 = 8.6023, and you need the calculator.

Why does standard deviation involve a square root?

Standard deviation is the square root of variance so the spread comes back in the same units as the data. Variance is an average of squared deviations, so it is in squared units. Taking the root turns "variance = 25 square inches" back into "standard deviation = 5 inches."

What is the distance between two points?

The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), the same Pythagorean idea. It is the core formula behind map apps, computer graphics and GPS distance checks. Square roots also turn up in finance (volatility is the square root of variance, scaled by time) and in physics (the speed of a falling object, the period of a pendulum).

Is there a difference between "square root" and "principal square root"?

Yes. Every positive number has two square roots, but the principal square root, which is what the √ sign and a calculator give by default, is the non-negative one. Distance and time problems only need the principal root because they cannot be negative. Solving x² = 25 in algebra needs both, x = 5 and x = −5.

Checking your answer

Square it. If you think √148 is 12.17, then 12.17 × 12.17 = 148.1089, close to 148, so the answer is about right. A root of a number above 1 is always smaller than the number, and a root of a number between 0 and 1 is bigger. A negative result from a positive input is wrong unless you are solving for both roots. Results here are rounded to four decimal places, so squaring a rounded decimal gives a number very close to the original, not always identical.