Square Root of 677

√677 at a glance

Exact value

√677

Decimal (10 places)

26.0192236625

Rounded

26.0 · 26.02 · 26.019

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.019224

Prime factorization

677

Cube root

8.780708

How to simplify √677

677 is a prime number, so its only factors are 1 and 677. There is no perfect-square factor to pull out, which means √677 is already in its simplest radical form.

The square root of any prime is irrational. If √677 were a fraction a/b in lowest terms, then a² = 677b², so 677 would divide a — and then 677 would divide b too, contradicting “lowest terms.” That is why the decimal 26.0192236625 is only a rounded value.

Where √677 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √677 lies between 26 and 27. 677 is 1 above 676 and 52 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.0189 (0% low)
• Tangent from 26, i.e. 26 + 1 ÷ 52: 26.0192 (0% high)
• Tangent from 27, i.e. 27 − 52 ÷ 54: 26.0370 (0.07% high)

For √677 the tangent at 26 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 677 is just 1 above 676.

Finding √677 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 677: following the tangent line down to zero simplifies to averaging x with 677 ÷ x.

Start from the nearest whole number, 26 (26² = 676):

Because the starting guess was already close, two steps are enough to match √677 = 26.0192236625 to every decimal shown.

√677 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √677 the pattern is [26; 52] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 677 is one more than a perfect square (26² + 1). A pattern that never ends is one more proof that √677 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 − 677y² = 1. Its smallest solution in positive whole numbers is x = 1,353, y = 52. Because the period is odd, the equation with −1 on the right also has a solution: 26² − 677 × 1² = −1.

√677 in geometry and everyday measurements

• 677 square feet is 62.9 m². Laid out as a square — a small house footprint or a lot — it is about 26.02 ft (26 ft) on a side.
• 677 = 1² + 26², so by the Pythagorean theorem √677 is the diagonal of a 1 × 26 rectangle — and the distance between the points (0, 0) and (1, 26) on a grid.

Square roots near √677 and related roots

• The cube root of 677 is about 8.780708.
• Squaring undoes the root: (√677)² = 677, while 677² = 458,329 — the number whose square root is 677.

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