√400 = 20, because 20 × 20 = 400
√400 = 20. 400 is a perfect square, so its root is the whole number 20: multiply 20 by itself and you get 400 exactly. Below is where that answer comes from, how to check it in one step, and how to estimate any square root without a calculator.
√400 = 20. The square root of a number is whatever you have to multiply by itself to get that number, and 20 × 20 = 400, so 20 is the answer.
400 is a perfect square — it is 20 squared — which means its root is a whole number with nothing left over. Strictly there are two numbers that square to 400, because −20 × −20 = 400 as well, but "the square root of 400" on its own means the positive one: 20.
In words: the square root of four hundred is twenty.
Squaring is the way back. 20 × 20 = 400, so the root is right. That check costs one multiplication and catches every slip.
It also gives you the fact in reverse: 20 squared is 400, written 20² = 400. Square roots and squares are the same fact read in opposite directions, exactly the way multiplication and division are.
Find the two square numbers it falls between and you have already bracketed the answer. 400 and 441 are 20² and 21², and 400 lies exactly on one of them.
To sharpen the estimate, see how far along the gap the number sits. From 400 to 441 is a jump of 41, and 400 is 0 of the way in, so the root lands roughly 20. 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 "√400 is a bit exactly 20" in a second is what keeps a longer calculation moving.
The square root of 400 is 20, because 20 × 20 = 400.
Yes. 400 is 20 squared, so its square root is the whole number 20.
20 × 20 = 400. 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 — 400 and 441 — then judge how far along the gap 400 sits. That gets you to roughly 20, which is close enough for almost any mental check.
The square numbers up to 25² each have a page here, and the square numbers index lists them all in order.