How to calculate a square root, step by step
To calculate a square root, ask what number multiplied by itself gives the number you started with. That number is the square root. The square root of 144 is 12, because 12 × 12 is 144. The square root of 100 is 10, because 10 × 10 is 100. And the square root of 25 is 5, because 5 × 5 is 25. Every square root is that question, and the methods below are just faster ways of answering it. Work it out in the square-root-calculator.
The worked example: the square root of 144
The worked example, step by step. Step 1 — ask what times itself is 144: 12 × 12 is 144, so the answer is 12. Step 2 — check for a perfect square: 144 is 12², so it is a perfect square and the answer is exact and whole. Step 3 — simplify the radical: there is nothing left to simplify, because the answer is already whole — the simplified form is 12. Step 4 — write the decimal: 12.000000 to six places. Step 5 — check it: 12 × 12 is 144, so the answer is right. If you square the answer and do not get 144, the answer is wrong — the check is the multiplication itself.
Perfect squares from memory
The fastest way to find a square root is to know the perfect squares by heart, the way you know the times tables. The first twelve are worth memorising: 1² is 1, 2² is 4, 3² is 9, 4² is 16, 5² is 25, 6² is 36, 7² is 49, 8² is 64, 9² is 81, 10² is 100, 11² is 121, and 12² is 144. If the number under the radical is one of these, the answer is immediate: the square root of 121 is 11, the square root of 81 is 9, and the square root of 144 is 12. The square-root-calculator shows whether any number you type is a perfect square.
Estimating between two perfect squares
When the number is not a perfect square, estimate between the two perfect squares it sits between. Take the square root of 20. The perfect squares around it are 16 and 25, so the answer is between 4 and 5. Because 20 is closer to 25 than to 16, the answer is closer to 5 — about 4.5. That is already close enough for most purposes, and it is a useful check on any calculator answer: if a calculator says the square root of 20 is far from 4.5, the number was mistyped.
The long-division method for hand calculation
When you need the digits, not just an estimate, the long-division method — the digit-by-digit method — finds them one at a time. Take the square root of 144. Step 1 — pair the digits from the right: 1 and 44. Step 2 — find the largest square that fits in the first pair, 1: that is 1², so the first digit of the answer is 1, and the remainder is 0. Step 3 — bring down the next pair, 44, and double the answer so far, 1, to get 2. Step 4 — find the digit that, put after the 2 and multiplied by itself, fits in 44: 2 followed by 2 is 22, and 22 × 2 is 44, so the next digit is 2. Step 5 — the remainder is 0, so the answer is complete: 12. The method works for any number, and it is how square roots were found before calculators — slow, but exact, and every digit it produces is a digit you can trust.
Simplifying radicals
Sometimes the exact answer is a radical, not a decimal, and simplifying it makes it easier to work with. Simplifying means pulling square factors out of the radical. Take the square root of 18. The number 18 is 9 × 2, and 9 is a perfect square, so the 3 comes out of the radical and the 2 stays in: the square root of 18 is 3√2. The rule is simple: split the number into a perfect square times whatever is left, take the square root of the perfect square, and leave the rest under the radical. If the number has no square factor — 2, 3, 7 — there is nothing to pull out, and the simplified form is the radical itself. The simplified radical is the exact answer; the decimal is only an approximation of it. The square-root-calculator shows the simplified radical for any number you type.
Checking by squaring the answer
Whatever method you used, check it by squaring the answer: multiply the answer by itself and see whether you get the number you started with. For a perfect square the check is exact: 12 × 12 is 144, so the square root of 144 is 12. For a number that is not a perfect square, the check is approximate at the places you kept: the square of the six-decimal answer should be the input to the places shown. If the square is nowhere near the input, the answer is wrong — and the most likely cause is a mistyped input, not a broken method. Run your numbers through the square-root-calculator and compare them with the worked example above.
Common mistakes
The first is taking the square root of each digit separately: the square root of 144 is not 1 and 2, it is 12. The second is forgetting that a square root has two answers, positive and negative: 12 and −12 both square to 144, and this calculator gives the positive one, the principal square root. The third is using a decimal where the exact radical belongs: for further working, carry the radical, not the rounded decimal, because the rounding compounds. The fourth is guessing without estimating first: a quick estimate between two perfect squares catches errors that no amount of careful long division will. And the fifth is skipping the check: one multiplication confirms the answer.
A quick check you can do in your head
You can sanity-check any square root without a calculator. First, square the answer roughly: if the answer is 12, 12 × 12 is 144, so an answer far from 12 for the square root of 144 is wrong. Second, use the perfect squares around it: the square root of a number between 121 and 144 must sit between 11 and 12. Third, check the last digit: a perfect square can only end in 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8 is never a perfect square. If a calculator says the square root of 144 is far from 12, the number is wrong — run your numbers through the square-root-calculator and compare them with the worked example above.