Square Root of 757

√757 at a glance

Exact value

√757

Decimal (10 places)

27.5136329844

Rounded

27.5 · 27.51 · 27.514

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.513633

Prime factorization

757

Cube root

9.113782

How to simplify √757

757 is a prime number, so its only factors are 1 and 757. There is no perfect-square factor to pull out, which means √757 is already in its simplest radical form.

The square root of any prime is irrational. If √757 were a fraction a/b in lowest terms, then a² = 757b², so 757 would divide a — and then 757 would divide b too, contradicting “lowest terms.” That is why the decimal 27.5136329844 is only a rounded value.

Where √757 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √757 lies between 27 and 28. 757 is 28 above 729 and 27 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.5091 (0.02% low)
• Tangent from 27, i.e. 27 + 28 ÷ 54: 27.5185 (0.02% high)
• Tangent from 28, i.e. 28 − 27 ÷ 56: 27.5179 (0.02% high)

For √757 the tangent at 28 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 757 is just 27 below 784.

Finding √757 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 757: following the tangent line down to zero simplifies to averaging x with 757 ÷ x.

Start from the nearest whole number, 28 (28² = 784):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √757 = 27.5136329844 to every decimal shown.

√757 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √757 the pattern is [27; 1, 1, 17, 1, 5, 5, 1, 17, 1, 1, 54] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √757 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 − 757y² = 1. Its smallest solution in positive whole numbers is x = 3,750,107,388,553, y = 136,299,971,388 — 13 digits for x, even though 757 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 1,369,326² − 757 × 49,769² = −1.

√757 in geometry and everyday measurements

• 757 square feet is 70.3 m². Laid out as a square — a small house footprint or a lot — it is about 27.51 ft (27 ft 6 in) on a side.
• 757 = 9² + 26², so by the Pythagorean theorem √757 is the diagonal of a 9 × 26 rectangle — and the distance between the points (0, 0) and (9, 26) on a grid.

Square roots near √757 and related roots

• The cube root of 757 is about 9.113782.
• Squaring undoes the root: (√757)² = 757, while 757² = 573,049 — the number whose square root is 757.

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