√169 = 13, because 13 × 13 = 169
√169 = 13. 169 is a perfect square, so its root is the whole number 13: multiply 13 by itself and you get 169 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.
√169 = 13. The square root of a number is whatever you have to multiply by itself to get that number, and 13 × 13 = 169, so 13 is the answer.
169 is a perfect square — it is 13 squared — which means its root is a whole number with nothing left over. Strictly there are two numbers that square to 169, because −13 × −13 = 169 as well, but "the square root of 169" on its own means the positive one: 13.
In words: the square root of one hundred and sixty-nine is thirteen.
Squaring is the way back. 13 × 13 = 169, so the root is right. That check costs one multiplication and catches every slip.
It also gives you the fact in reverse: 13 squared is 169, written 13² = 169. 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. 169 and 196 are 13² and 14², and 169 lies exactly on one of them.
To sharpen the estimate, see how far along the gap the number sits. From 169 to 196 is a jump of 27, and 169 is 0 of the way in, so the root lands roughly 13. 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 "√169 is a bit exactly 13" in a second is what keeps a longer calculation moving.
The square root of 169 is 13, because 13 × 13 = 169.
Yes. 169 is 13 squared, so its square root is the whole number 13.
13 × 13 = 169. 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 — 169 and 196 — then judge how far along the gap 169 sits. That gets you to roughly 13, 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.