√242 is irrational — 15.5563 to four decimal places, 15.56 to two
√242 ≈ 15.5563. 242 is not a perfect square, so the root is an irrational number and 15.5563 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.
√242 ≈ 15.5563. Rounded to two decimal places that is 15.56.
242 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 15 and 16: 15 × 15 = 225 and 16 × 16 = 256, and 242 sits between those two, so √242 sits between 15 and 16.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 15.5563 is a rounded value, not the exact one — the exact answer can only be written as √242 itself, or in simplified form as 11√2.
Square your answer and see how close it lands. 15.56 × 15.56 = 242.1136, which is within a whisker of 242 — 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 242. That gap is the rounding, not an error in the root.
√242 simplifies to 11√2, because 242 = 121 × 2 and 121 is a perfect square. Pulling the 11 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 11 × √2 = 11 × 1.4142 = 15.5563, the same value as √242. 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. 225 and 256 are 15² and 16², and 242 lies between them, so the root is between 15 and 16.
To sharpen the estimate, see how far along the gap the number sits. From 225 to 256 is a jump of 31, and 242 is 17 of the way in, so the root lands roughly 15.55. 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 "√242 is a bit over 15" in a second is what keeps a longer calculation moving.
The square root of 242 is approximately 15.5563, or 15.56 to two decimal places. It is an irrational number, so the decimal does not end.
No. 242 falls between 225 (15²) and 256 (16²), so its square root is not a whole number.
15 × 15 = 225. 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 — 225 and 256 — then judge how far along the gap 242 sits. That gets you to roughly 15.55, which is close enough for almost any mental check.
√242 = 11√2, because 242 = 121 × 2 and 121 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.