To find the square root of a number, you’re looking for a value that, when multiplied by itself , gives the original number. For example, the square root of 25 is 5 because 5×5=255\times 5=255×5=25.

Below is a friendly, step‑by‑step guide with several methods you can use, from quick mental tricks to precise hand‑calculation and calculators.

What a square root really is

  • The square of a number is the result of multiplying it by itself (like 7×7=497\times 7=497×7=49).
  • The square root reverses that: 49=7\sqrt{49}=749​=7.
  • In general, x=x1/2\sqrt{x}=x^{1/2}x​=x1/2.

Think of a square root as: “What side length does a square need so that its area is this number?”

Fast ways for perfect squares

Perfect squares are numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.

1. Using memorized squares

If the number is small and common, you can often just recall:

  • 4=2\sqrt{4}=24​=2, 9=3\sqrt{9}=39​=3, 16=4\sqrt{16}=416​=4.
  • 25=5\sqrt{25}=525​=5, 36=6\sqrt{36}=636​=6, 49=7\sqrt{49}=749​=7, 64=8\sqrt{64}=864​=8, 81=9\sqrt{81}=981​=9, 100=10\sqrt{100}=10100​=10.

This is the “mental math” baseline almost everyone uses.

2. Prime factorization method (for perfect squares)

This is a clean, systematic method taught in many school texts.

Idea:
Break the number into prime factors, pair them, then take one from each pair and multiply. Steps (example: 144\sqrt{144}144​)

  1. Factor into primes:
    144=2×2×2×2×3×3144=2\times 2\times 2\times 2\times 3\times 3144=2×2×2×2×3×3.

  2. Group equal factors into pairs:
    (2×2)⋅(2×2)⋅(3×3)(2\times 2)\cdot (2\times 2)\cdot (3\times 3)(2×2)⋅(2×2)⋅(3×3).

  3. Take one from each pair: 2,2,32,2,32,2,3.

  4. Multiply them: 2×2×3=122\times 2\times 3=122×2×3=12.

So 144=12\sqrt{144}=12144​=12.

This works best when the number is an exact square and factoring is easy.

3. Repeated subtraction of odd numbers (for perfect squares)

This is more of a “math trick,” useful for understanding, not speed.

Fact: If you subtract consecutive odd numbers from a perfect square, you reach 0 after exactly as many steps as its square root. Example for 16:

  • 16−1=1516-1=1516−1=15
  • 15−3=1215-3=1215−3=12
  • 12−5=712-5=712−5=7
  • 7−7=07-7=07−7=0

You subtracted 4 odd numbers, so 16=4\sqrt{16}=416​=4.

Long division–style method (for any number)

This is the “by hand” algorithm that works for non‑perfect squares (like 180, 2, or 10) and gives as many decimal places as you want.

I’ll use 180\sqrt{180}180​ as an example.

Step 1: Group digits in pairs

  • Starting from the decimal point and moving left and right, group digits in pairs.
  • For 180: write it as 1 | 80.

If you had decimals, like 180.25, it would be 1 | 80 . 25 (pairs on both sides).

Step 2: First digit of the root

  • Find the largest integer whose square is ≤ the first group (1).
  • 12=11^2=112=1, 22=4>12^2=4>122=4>1, so choose 1.
  • Write 1 as the first digit of the root; subtract 12=11^2=112=1 from 1, remainder 0.

Root so far: 1.

Step 3: Bring down next pair

  • Bring down the next pair 80, making the current “dividend” 80.
  • Double the current root: 1→21\to 21→2. This becomes the “trial divisor base.”

Step 4: Find the next digit

We now look for a digit xxx such that:

  • (20+x)×x≤80(20+x)\times x\le 80(20+x)×x≤80.

Try digits:

  • x=3:(20+3)×3=23×3=69≤80x=3:(20+3)\times 3=23\times 3=69\le 80x=3:(20+3)×3=23×3=69≤80.
  • x=4:(20+4)×4=24×4=96>80x=4:(20+4)\times 4=24\times 4=96>80x=4:(20+4)×4=24×4=96>80.

So x=3x=3x=3 works.

  • Add 3 to the root: it becomes 13.
  • Subtract 696969 from 80: remainder 11.

Step 5: Get decimal places

To continue beyond the integer part:

  1. Put a decimal point in the root (13.) and in the number (180.00…).
  2. Bring down a pair of zeros → remainder becomes 1100.
  3. Double the current root (13 → 26), and repeat the “guess digit” step with (260+x)×x≤1100(260+x)\times x\le 1100(260+x)×x≤1100.

Carrying this out gives 180≈13.4\sqrt{180}\approx 13.4180​≈13.4, and more digits come by repeating the same steps.

This is the classic “digit‑by‑digit square root algorithm” also discussed in modern tutorials and videos as a historical algorithm for exams and manual calculations.

Estimation and approximation (quick mental method)

When you don’t need exact digits, just a good approximation, use nearby perfect squares.

Example: 50\sqrt{50}50​.

  1. Notice 72=497^2=4972=49 and 82=648^2=6482=64, so 50\sqrt{50}50​ is a bit more than 7.
  1. Because 50 is very close to 49, a rough estimate is about 7.1.

For a slightly more refined mental estimate, people often use linear approximations or start with a nearby perfect square and iteratively correct the guess (like Newton’s method).

Example: 10\sqrt{10}10​.

  • 32=93^2=932=9, 42=164^2=1642=16, so 10\sqrt{10}10​ is between 3 and 4.
  • Since 10 is closer to 9 than to 16, it’s a bit above 3: roughly 3.16 (the true value is about 3.162…).

On forums, this kind of “speedy square root” or mental trick is a common topic, often linked to Newton’s method or approximations around known squares.

Using a calculator

The most practical route in everyday life is just to use the square root function.

  • Type the number (e.g., 81).
  • Press the x\sqrt{\phantom{x}}x​ key, or choose “sqrt” in a scientific calculator app.
  • The display gives the square root (for 81, that’s 9).

There are also dedicated “square root calculator” websites and apps where you just enter any positive number and get the square root instantly.

Different methods side‑by‑side

[3][1] [1] [3][1] [3][1] [1] [1] [10][1] [1] [5][3][1] [5][3] [8][1] [8]
Method Best for Key idea Example result
Memorized squares Small perfect squares Recall that e.g. 9 = 3×3, so √9 = 3.√81 = 9.
Prime factorization Exact roots for perfect squares Factor into primes, pair equal factors, multiply one from each pair.√144 = 12.
Repeated subtraction Conceptual understanding Subtract consecutive odd numbers until 0; number of steps is the root.16 → 4 subtractions → √16 = 4.
Long division algorithm Any number, many decimals Group digits, grow the root digit by digit with a trial divisor.√180 ≈ 13.4.
Estimation/approximation Quick mental rough value Use nearby perfect squares and adjust.√50 ≈ 7.1; √10 ≈ 3.16.
Calculator / online tool Fast, everyday use Enter number and press √ or use online calculator.√2 ≈ 1.4142.

Tiny story to remember it

Imagine you’re designing square tiles for a game. You know each tile’s area , say 49 square units, but you need the side length so characters fit just right. You ask, “What number times itself gives 49?” That question is exactly “What is the square root of 49?”—and the answer, 7, is the side length you should use.

Quick recap (TL;DR)

  • A square root is the number that squares to your original number, x=x1/2\sqrt{x}=x^{1/2}x​=x1/2.
  • For neat numbers (perfect squares), memorize or use prime factorization.
  • For any number, you can use the long division algorithm or a calculator.
  • For fast mental answers, estimate using nearby perfect squares and simple approximations.

Information gathered from public forums or data available on the internet and portrayed here.