√361 = 19, because 19 × 19 = 361
√361 = 19. 361 is a perfect square, so its root is the whole number 19: multiply 19 by itself and you get 361 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.
√361 = 19. The square root of a number is whatever you have to multiply by itself to get that number, and 19 × 19 = 361, so 19 is the answer.
361 is a perfect square — it is 19 squared — which means its root is a whole number with nothing left over. Strictly there are two numbers that square to 361, because −19 × −19 = 361 as well, but "the square root of 361" on its own means the positive one: 19.
In words: the square root of three hundred and sixty-one is nineteen.
Squaring is the way back. 19 × 19 = 361, so the root is right. That check costs one multiplication and catches every slip.
It also gives you the fact in reverse: 19 squared is 361, written 19² = 361. 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. 361 and 400 are 19² and 20², and 361 lies exactly on one of them.
To sharpen the estimate, see how far along the gap the number sits. From 361 to 400 is a jump of 39, and 361 is 0 of the way in, so the root lands roughly 19. 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 "√361 is a bit exactly 19" in a second is what keeps a longer calculation moving.
The square root of 361 is 19, because 19 × 19 = 361.
Yes. 361 is 19 squared, so its square root is the whole number 19.
19 × 19 = 361. 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 — 361 and 400 — then judge how far along the gap 361 sits. That gets you to roughly 19, 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.