How to Calculate a Square Root

What's the square root of 1,500? Without a calculator, most people would guess somewhere around 38 or 39 and stop there.

With a method nearly 4,000 years old, you can get 38.7298 in just three simple steps of arithmetic. The same method is essentially what computers use today.

Here are four ways to calculate square roots, from instant recognition to precise hand calculation.

A Rough Guess Is the Fastest Path to a Precise Answer

It seems like precision should require a careful, slow procedure. The Babylonian method works the other way: start with a rough guess, and a simple averaging step roughly doubles the number of correct digits each time. A poor starting guess costs only an extra step or two.

What Is a Square Root?

The square root of a number x is a value that, multiplied by itself, gives x:

√49 = 7   because 7 × 7 = 49

Every positive number has two square roots, one positive and one negative (7 and −7), but the symbol √ means the positive one, called the principal square root.

Method 1: Recognize Perfect Squares

A perfect square is a whole number squared. Memorizing the first 20 or so makes many roots instant:

A handy trick: √(a × 100) = √a × 10. So √2500 = √25 × 10 = 50. See the full square numbers list.

Method 2: Estimate Between Perfect Squares

For numbers that aren't perfect squares, find the squares on either side.

Example: √50

7² = 49 and 8² = 64, so √50 is between 7 and 8
50 is just 1 above 49, out of a gap of 15
√50 ≈ 7 + 1/15 ≈ 7.07

The true value is 7.0711. This linear estimate works well close to a perfect square.

Method 3: Simplify Using Prime Factors

To write a square root in exact form, pull out perfect square factors:

Example: √72

Step 1: Factor 72: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². Use the prime factorization of 72.

Step 2: Pair up factors. Each pair comes out of the root as one factor:

√72 = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2

Since √2 ≈ 1.41421, √72 ≈ 6 × 1.41421 ≈ 8.485.

This works because √(a × b) = √a × √b for non-negative numbers.

Method 4: The Babylonian Method (Precise)

To find √S:

1. Make a guess x
2. Compute S ÷ x
3. Average them: x_new = (x + S ÷ x) ÷ 2
4. Repeat until the value stops changing

Example: √1500

Start with 40 (since 40² = 1600, close to 1500):

After just three steps, you have 38.729833, matching the true value to six decimal places.

Why it works: if x is too large, S/x is too small, and vice versa. The true root always lies between them, so their average is closer. Try it with the scientific calculator, and check results in the square roots list.

An Insider Reference: The √ Symbol

The Babylonian clay tablet YBC 7289, from around 1800–1600 BC, shows √2 approximated to about six decimal places, likely by a method like the one above.

The √ symbol itself is much younger. It first appeared in print in 1525, in Die Coss, an algebra book by German mathematician Christoph Rudolff. It's thought to derive from a stylized lowercase r, for the Latin radix, meaning "root." That's also where the word radical comes from. René Descartes later added the horizontal bar, the vinculum, over the number in 1637.

Square Roots of Non-Perfect Squares Are Irrational

If a whole number isn't a perfect square, its square root is irrational: its decimal never ends and never repeats. √2 = 1.41421356… will never settle into a pattern. That's why simplified radical form (like 6√2) is often preferred for exact answers. See Why Some Numbers Cannot Be Written as Fractions.

Two Concepts Worth Knowing

Radical

A radical is an expression with a root symbol. "Simplest radical form" means no perfect square factors remain under the √: √72 becomes 6√2.

Newton's Method

The Babylonian method is a special case of Newton's method, a general technique for finding roots of equations using tangent lines. See How Computers Actually Calculate Square Roots.

Quick Answer: How Do You Calculate a Square Root?

For perfect squares, recall which number times itself gives the value (√144 = 12). To estimate, find the perfect squares on either side. To simplify, factor out perfect squares (√72 = 6√2). For precision, use the Babylonian method: guess x, then repeat x = (x + S/x) ÷ 2. Starting from 40, √1500 ≈ 38.7298 after three steps.

Try Them Yourself

• Square Roots List: check any square root
• Square Numbers List: memorize perfect squares
• Prime Factorization of 72: simplify radicals
• Scientific Calculator: run the Babylonian method
• Why You Can't Take the Square Root of a Negative Number — Or Can You?: what happens below zero
• Why Does the Pythagorean Theorem Work?: where square roots show up most

Estimate √200 by hand using Method 2, then refine it with two Babylonian steps. Compare with the square roots list and count your correct digits.