√512 is irrational — 22.6274 to four decimal places, 22.63 to two
√512 ≈ 22.6274. Five hundred and twelve is a power of two, so its root has a tidy exact form that also explains the shape of A4 paper.
√512 ≈ 22.6274. Rounded to two decimal places that is 22.63.
512 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 512 sits between those two, so √512 sits between 22 and 23.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 22.6274 is a rounded value, not the exact one — the exact answer can only be written as √512 itself, or in simplified form as 16√2.
Square your answer and see how close it lands. 22.63 × 22.63 = 512.1169, which is within a whisker of 512 — close enough to confirm the root without a calculator agreeing digit for digit.
√512 simplifies to 16√2, because 512 = 256 × 2 and 256 is a perfect square. Pulling the 16 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 16 × √2 = 16 × 1.4142 = 22.6274, the same value as √512. Simplified form is the same number written more tidily, not a different one.
512 is 2 multiplied by itself nine times. Taking a square root halves the power, so √512 = 2 to the power 4.5 = 16 × √2 = 16√2 ≈ 22.6274.
The same √2 is the ratio of the long side to the short side of every ISO paper size. Fold A4 in half and you get A5 with the same shape — only possible because the sides are in the ratio √2 : 1.
As a check: 22² = 484 and 23² = 529, so √512 has to be between 22 and 23, closer to 23.
512 lies between 484 (22²) and 529 (23²), 28 of the way along a gap of 45, so a quick linear estimate is 22.62. Bracketing between the nearest squares is the fastest mental check for any root.
The square root of 512 is approximately 22.6274, or 22.63 to two decimal places. It is an irrational number, so the decimal does not end.
No. 512 falls between 484 (22²) and 529 (23²), so its square root is not a whole number.
Bracket it between the two nearest square numbers — 484 and 529 — then judge how far along the gap 512 sits. That gets you to roughly 22.62, which is close enough for almost any mental check.
8, because 8 × 8 × 8 = 512. 512 is a perfect cube and a power of two (2⁹), but not a perfect square.
No. 22² = 484 and 23² = 529, so √512 ≈ 22.627 lies between them. Its exact form is 16√2.
√512 = 16√2, because 512 = 256 × 2 and 256 is a perfect square.
Every square number up to 25² — and the roots people most often look up — is listed on the square numbers page.