√60 is irrational — 7.746 to four decimal places, 7.75 to two
√60 ≈ 7.746. Sixty is not a perfect square, but a four-thousand-year-old method gets its root to four decimal places in one step.
√60 ≈ 7.746. Rounded to two decimal places that is 7.75.
60 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 7 and 8: 7 × 7 = 49 and 8 × 8 = 64, and 60 sits between those two, so √60 sits between 7 and 8.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 7.746 is a rounded value, not the exact one — the exact answer can only be written as √60 itself, or in simplified form as 2√15.
Square your answer and see how close it lands. 7.75 × 7.75 = 60.0625, which is within a whisker of 60 — close enough to confirm the root without a calculator agreeing digit for digit.
√60 simplifies to 2√15, because 60 = 4 × 15 and 4 is a perfect square. Pulling the 2 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 2 × √15 = 2 × 3.873 = 7.746, the same value as √60. Simplified form is the same number written more tidily, not a different one.
60 lies between 49 (7²) and 64 (8²), eleven-fifteenths of the way along, so a first guess is about 7.73. To sharpen it, use the Babylonian method: average your guess with 60 divided by your guess. (7.75 + 60 ÷ 7.75) ÷ 2 = (7.75 + 7.7419) ÷ 2 ≈ 7.7460 — correct to four decimal places after one step.
The method is named for the Babylonians, who worked in base 60 and computed square roots with remarkable accuracy; a clay tablet known as YBC 7289, from around 1800–1600 BC, gives √2 to the equivalent of six decimal places.
In exact form, √60 = 2√15, because 60 = 4 × 15 and the 4 comes out from under the root as 2.
60 lies between 49 (7²) and 64 (8²), 11 of the way along a gap of 15, so a quick linear estimate is 7.73. Bracketing between the nearest squares is the fastest mental check for any root.
The square root of 60 is approximately 7.746, or 7.75 to two decimal places. It is an irrational number, so the decimal does not end.
No. 60 falls between 49 (7²) and 64 (8²), so its square root is not a whole number.
Bracket it between the two nearest square numbers — 49 and 64 — then judge how far along the gap 60 sits. That gets you to roughly 7.73, which is close enough for almost any mental check.
60² = 60 × 60 = 3,600. That is the opposite operation: squaring 60 multiplies it by itself, while √60 ≈ 7.746 asks what number multiplied by itself gives 60.
√60 = 2√15, because 60 = 4 × 15 and 4 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.