√500 is irrational — 22.3607 to four decimal places, 22.36 to two
√500 ≈ 22.3607. 500 is not a perfect square, so the root is an irrational number and 22.3607 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.
√500 ≈ 22.3607. Rounded to two decimal places that is 22.36.
500 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 22 and 23: 22 × 22 = 484 and 23 × 23 = 529, and 500 sits between those two, so √500 sits between 22 and 23.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 22.3607 is a rounded value, not the exact one — the exact answer can only be written as √500 itself, or in simplified form as 10√5.
Square your answer and see how close it lands. 22.36 × 22.36 = 499.9696, which is within a whisker of 500 — 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 500. That gap is the rounding, not an error in the root.
√500 simplifies to 10√5, because 500 = 100 × 5 and 100 is a perfect square. Pulling the 10 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 10 × √5 = 10 × 2.2361 = 22.3607, the same value as √500. 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. 484 and 529 are 22² and 23², and 500 lies between them, so the root is between 22 and 23.
To sharpen the estimate, see how far along the gap the number sits. From 484 to 529 is a jump of 45, and 500 is 16 of the way in, so the root lands roughly 22.36. 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 "√500 is a bit over 22" in a second is what keeps a longer calculation moving.
The square root of 500 is approximately 22.3607, or 22.36 to two decimal places. It is an irrational number, so the decimal does not end.
No. 500 falls between 484 (22²) and 529 (23²), so its square root is not a whole number.
22 × 22 = 484. 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 — 484 and 529 — then judge how far along the gap 500 sits. That gets you to roughly 22.36, which is close enough for almost any mental check.
√500 = 10√5, because 500 = 100 × 5 and 100 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.