√288 is irrational — 16.9706 to four decimal places, 16.97 to two
√288 ≈ 16.9706. 288 is not a perfect square, so the root is an irrational number and 16.9706 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.
√288 ≈ 16.9706. Rounded to two decimal places that is 16.97.
288 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 16 and 17: 16 × 16 = 256 and 17 × 17 = 289, and 288 sits between those two, so √288 sits between 16 and 17.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 16.9706 is a rounded value, not the exact one — the exact answer can only be written as √288 itself, or in simplified form as 12√2.
Square your answer and see how close it lands. 16.97 × 16.97 = 287.9809, which is within a whisker of 288 — 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 288. That gap is the rounding, not an error in the root.
√288 simplifies to 12√2, because 288 = 144 × 2 and 144 is a perfect square. Pulling the 12 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 12 × √2 = 12 × 1.4142 = 16.9706, the same value as √288. 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. 256 and 289 are 16² and 17², and 288 lies between them, so the root is between 16 and 17.
To sharpen the estimate, see how far along the gap the number sits. From 256 to 289 is a jump of 33, and 288 is 32 of the way in, so the root lands roughly 16.97. 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 "√288 is a bit over 16" in a second is what keeps a longer calculation moving.
The square root of 288 is approximately 16.9706, or 16.97 to two decimal places. It is an irrational number, so the decimal does not end.
No. 288 falls between 256 (16²) and 289 (17²), so its square root is not a whole number.
16 × 16 = 256. 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 — 256 and 289 — then judge how far along the gap 288 sits. That gets you to roughly 16.97, which is close enough for almost any mental check.
√288 = 12√2, because 288 = 144 × 2 and 144 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.