√800 is irrational — 28.2843 to four decimal places, 28.28 to two
√800 ≈ 28.2843. 800 is not a perfect square, so the root is an irrational number and 28.2843 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.
√800 ≈ 28.2843. Rounded to two decimal places that is 28.28.
800 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 28 and 29: 28 × 28 = 784 and 29 × 29 = 841, and 800 sits between those two, so √800 sits between 28 and 29.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 28.2843 is a rounded value, not the exact one — the exact answer can only be written as √800 itself, or in simplified form as 20√2.
Square your answer and see how close it lands. 28.28 × 28.28 = 799.7584, which is within a whisker of 800 — 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 800. That gap is the rounding, not an error in the root.
√800 simplifies to 20√2, because 800 = 400 × 2 and 400 is a perfect square. Pulling the 20 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 20 × √2 = 20 × 1.4142 = 28.2843, the same value as √800. 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. 784 and 841 are 28² and 29², and 800 lies between them, so the root is between 28 and 29.
To sharpen the estimate, see how far along the gap the number sits. From 784 to 841 is a jump of 57, and 800 is 16 of the way in, so the root lands roughly 28.28. 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 "√800 is a bit over 28" in a second is what keeps a longer calculation moving.
The square root of 800 is approximately 28.2843, or 28.28 to two decimal places. It is an irrational number, so the decimal does not end.
No. 800 falls between 784 (28²) and 841 (29²), so its square root is not a whole number.
28 × 28 = 784. 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 — 784 and 841 — then judge how far along the gap 800 sits. That gets you to roughly 28.28, which is close enough for almost any mental check.
√800 = 20√2, because 800 = 400 × 2 and 400 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.