Square Root of 775

√775 at a glance

Exact value

5√31

Decimal (10 places)

27.8388218142

Rounded

27.8 · 27.84 · 27.839

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.838822

Prime factorization

5² × 31

Cube root

9.185453

How to simplify √775

Look for the largest perfect square that divides 775. Here it is 25 (5²), because 775 = 25 × 31 and 31 has no square factor left:

The prime factorization tells the same story: 775 = 5² × 31. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 31 stays inside.

Check: (5√31)² = 5² × 31 = 25 × 31 = 775. As a decimal, 5√31 = 5 × 5.5677643628 ≈ 27.8388218142.

Where √775 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √775 lies between 27 and 28. 775 is 46 above 729 and 9 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.8364 (0.01% low)
• Tangent from 27, i.e. 27 + 46 ÷ 54: 27.8519 (0.05% high)
• Tangent from 28, i.e. 28 − 9 ÷ 56: 27.8393 (0% high)

For √775 the tangent at 28 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 775 is just 9 below 784.

Finding √775 with the Babylonian method

If a guess is too big, 775 ÷ 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√775) in one step.

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 √775 = 27.8388218142 to every decimal shown.

√775 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √775 the pattern is [27; 1, 5, 4, 1, 8, 2, 8, 1, 4, 5, 1, 54] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √775 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 − 775y² = 1. Its smallest solution in positive whole numbers is x = 4,620,799, y = 165,984.

√775 in geometry and everyday measurements

• 775 square feet is 72 m². Laid out as a square — a small house footprint or a lot — it is about 27.84 ft (27 ft 10 in) on a side.
• 775 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √775 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 √775 as its space diagonal.
• Since √775 = 5√31, a length of √775 is exactly 5 copies of the length √31 laid end to end.

Square roots near √775 and related roots

• The cube root of 775 is about 9.185453.
• Squaring undoes the root: (√775)² = 775, while 775² = 600,625 — the number whose square root is 775.

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