√450 is irrational — 21.2132 to four decimal places, 21.21 to two
√450 ≈ 21.2132. 450 is not a perfect square, so the root is an irrational number and 21.2132 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.
√450 ≈ 21.2132. Rounded to two decimal places that is 21.21.
450 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 21 and 22: 21 × 21 = 441 and 22 × 22 = 484, and 450 sits between those two, so √450 sits between 21 and 22.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 21.2132 is a rounded value, not the exact one — the exact answer can only be written as √450 itself, or in simplified form as 15√2.
Square your answer and see how close it lands. 21.21 × 21.21 = 449.8641, which is within a whisker of 450 — 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 450. That gap is the rounding, not an error in the root.
√450 simplifies to 15√2, because 450 = 225 × 2 and 225 is a perfect square. Pulling the 15 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 15 × √2 = 15 × 1.4142 = 21.2132, the same value as √450. 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. 441 and 484 are 21² and 22², and 450 lies between them, so the root is between 21 and 22.
To sharpen the estimate, see how far along the gap the number sits. From 441 to 484 is a jump of 43, and 450 is 9 of the way in, so the root lands roughly 21.21. 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 "√450 is a bit over 21" in a second is what keeps a longer calculation moving.
The square root of 450 is approximately 21.2132, or 21.21 to two decimal places. It is an irrational number, so the decimal does not end.
No. 450 falls between 441 (21²) and 484 (22²), so its square root is not a whole number.
21 × 21 = 441. 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 — 441 and 484 — then judge how far along the gap 450 sits. That gets you to roughly 21.21, which is close enough for almost any mental check.
√450 = 15√2, because 450 = 225 × 2 and 225 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.