Square Root of 773

√773 at a glance

Exact value

√773

Decimal (10 places)

27.8028775489

Rounded

27.8 · 27.80 · 27.803

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.802878

Prime factorization

773

Cube root

9.177544

How to simplify √773

773 is a prime number, so its only factors are 1 and 773. There is no perfect-square factor to pull out, which means √773 is already in its simplest radical form.

The square root of any prime is irrational. If √773 were a fraction a/b in lowest terms, then a² = 773b², so 773 would divide a — and then 773 would divide b too, contradicting “lowest terms.” That is why the decimal 27.8028775489 is only a rounded value.

Where √773 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √773 lies between 27 and 28. 773 is 44 above 729 and 11 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.8000 (0.01% low)
• Tangent from 27, i.e. 27 + 44 ÷ 54: 27.8148 (0.04% high)
• Tangent from 28, i.e. 28 − 11 ÷ 56: 27.8036 (0% high)

For √773 the tangent at 28 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 773 is just 11 below 784.

Finding √773 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 773: following the tangent line down to zero simplifies to averaging x with 773 ÷ x.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √773 = 27.8028775489 to every decimal shown.

√773 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √773 the pattern is [27; 1, 4, 13, 1, 2, 2, 1, 13, 4, 1, 54] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √773 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 − 773y² = 1. Its smallest solution in positive whole numbers is x = 3,607,394,696,649, y = 129,748,968,980 — 13 digits for x, even though 773 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: 1,343,018² − 773 × 48,305² = −1.

√773 in geometry and everyday measurements

• 773 square feet is 71.8 m². Laid out as a square — a small house footprint or a lot — it is about 27.8 ft (27 ft 10 in) on a side.
• 773 = 17² + 22², so by the Pythagorean theorem √773 is the diagonal of a 17 × 22 rectangle — and the distance between the points (0, 0) and (17, 22) on a grid.

Square roots near √773 and related roots

• The cube root of 773 is about 9.177544.
• Squaring undoes the root: (√773)² = 773, while 773² = 597,529 — the number whose square root is 773.

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