√40 is irrational — 6.3246 to four decimal places, 6.32 to two
√40 ≈ 6.3246. 40 is not a perfect square, so the root is an irrational number and 6.3246 is a rounded value. Below is where that answer comes from, how to check it in one step, and how to estimate any square root without a calculator.
√40 ≈ 6.3246. Rounded to two decimal places that is 6.32.
40 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 6 and 7: 6 × 6 = 36 and 7 × 7 = 49, and 40 sits between those two, so √40 sits between 6 and 7.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 6.3246 is a rounded value, not the exact one — the exact answer can only be written as √40 itself, or in simplified form as 2√10.
Square your answer and see how close it lands. 6.32 × 6.32 = 39.9424, which is within a whisker of 40 — close enough to confirm the root without a calculator agreeing digit for digit.
Because rounding happens at the end, squaring a rounded root will never give back exactly 40. That gap is the rounding, not an error in the root.
√40 simplifies to 2√10, because 40 = 4 × 10 and 4 is a perfect square. Pulling the 2 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 2 × √10 = 2 × 3.1623 = 6.3246, the same value as √40. Simplified form is the same number written more tidily, not a different one.
Find the two square numbers it falls between and you have already bracketed the answer. 36 and 49 are 6² and 7², and 40 lies between them, so the root is between 6 and 7.
To sharpen the estimate, see how far along the gap the number sits. From 36 to 49 is a jump of 13, and 40 is 4 of the way in, so the root lands roughly 6.31. That linear guess is usually right to within a tenth, which is plenty for a sanity check.
Knowing the square numbers up to 25² by heart is what makes this instant. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576 and 625.
Square roots turn up any time an area, a distance or a spread has to be turned back into a length. The diagonal of a rectangle, the standard deviation of a set of numbers, the distance between two points on a grid — all of them finish with a square root.
That is why estimating one quickly is worth more than memorising a decimal. Being able to say "√40 is a bit over 6" in a second is what keeps a longer calculation moving.
The square root of 40 is approximately 6.3246, or 6.32 to two decimal places. It is an irrational number, so the decimal does not end.
No. 40 falls between 36 (6²) and 49 (7²), so its square root is not a whole number.
6 × 6 = 36. Squaring is the inverse of taking a square root, so it is also the way to check a root you have just worked out.
Bracket it between the two nearest square numbers — 36 and 49 — then judge how far along the gap 40 sits. That gets you to roughly 6.31, which is close enough for almost any mental check.
√40 = 2√10, because 40 = 4 × 10 and 4 is a perfect square.
The square numbers up to 25² each have a page here, and the square numbers index lists them all in order.