√150 is irrational — 12.2474 to four decimal places, 12.25 to two
√150 ≈ 12.2474. 150 is not a perfect square, so the root is an irrational number and 12.2474 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.
√150 ≈ 12.2474. Rounded to two decimal places that is 12.25.
150 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 12 and 13: 12 × 12 = 144 and 13 × 13 = 169, and 150 sits between those two, so √150 sits between 12 and 13.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 12.2474 is a rounded value, not the exact one — the exact answer can only be written as √150 itself, or in simplified form as 5√6.
Square your answer and see how close it lands. 12.25 × 12.25 = 150.0625, which is within a whisker of 150 — 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 150. That gap is the rounding, not an error in the root.
√150 simplifies to 5√6, because 150 = 25 × 6 and 25 is a perfect square. Pulling the 5 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 5 × √6 = 5 × 2.4495 = 12.2474, the same value as √150. 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. 144 and 169 are 12² and 13², and 150 lies between them, so the root is between 12 and 13.
To sharpen the estimate, see how far along the gap the number sits. From 144 to 169 is a jump of 25, and 150 is 6 of the way in, so the root lands roughly 12.24. 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 "√150 is a bit over 12" in a second is what keeps a longer calculation moving.
The square root of 150 is approximately 12.2474, or 12.25 to two decimal places. It is an irrational number, so the decimal does not end.
No. 150 falls between 144 (12²) and 169 (13²), so its square root is not a whole number.
12 × 12 = 144. 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 — 144 and 169 — then judge how far along the gap 150 sits. That gets you to roughly 12.24, which is close enough for almost any mental check.
√150 = 5√6, because 150 = 25 × 6 and 25 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.