Square Root of 751

√751 at a glance

Exact value

√751

Decimal (10 places)

27.4043792121

Rounded

27.4 · 27.40 · 27.404

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.404379

Prime factorization

751

Cube root

9.089639

How to simplify √751

751 is a prime number, so its only factors are 1 and 751. There is no perfect-square factor to pull out, which means √751 is already in its simplest radical form.

The square root of any prime is irrational. If √751 were a fraction a/b in lowest terms, then a² = 751b², so 751 would divide a — and then 751 would divide b too, contradicting “lowest terms.” That is why the decimal 27.4043792121 is only a rounded value.

Where √751 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √751 lies between 27 and 28. 751 is 22 above 729 and 33 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.4000 (0.02% low)
• Tangent from 27, i.e. 27 + 22 ÷ 54: 27.4074 (0.01% high)
• Tangent from 28, i.e. 28 − 33 ÷ 56: 27.4107 (0.02% high)

For √751 the tangent at 27 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 751 is just 22 above 729.

Finding √751 with the Babylonian method

If a guess is too big, 751 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√751) in one step.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √751 = 27.4043792121 to every decimal shown.

√751 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √751 the pattern is [27; 2, 2, 8, 1, 2, 1, 3, 5, 1, 4, 1, 1, …] with the block of 52 terms after the semicolon repeating forever (only the first 12 of the 52 are shown). A pattern that never ends is one more proof that √751 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 751y² = 1. Its smallest solution in positive whole numbers is x = 7,293,318,466,794,882,424,418,960, y = 266,136,970,677,206,024,456,793 — 25 digits for x, even though 751 is small, which is what makes Pell’s equation famous.

√751 in geometry and everyday measurements

• 751 square feet is 69.8 m². Laid out as a square — a small house footprint or a lot — it is about 27.4 ft (27 ft 5 in) on a side.
• 751 is not a sum of two whole-number squares — 751 is itself a prime that is one less than a multiple of 4, which rules that out — so √751 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √751 as its space diagonal.

Square roots near √751 and related roots

• The cube root of 751 is about 9.089639.
• Squaring undoes the root: (√751)² = 751, while 751² = 564,001 — the number whose square root is 751.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.