Square Root of 649

√649 at a glance

Exact value

√649

Decimal (10 places)

25.4754784057

Rounded

25.5 · 25.48 · 25.475

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.475478

Prime factorization

11 × 59

Cube root

8.657947

How to simplify √649

The prime factorization of 649 is 11 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √649 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 649, 11 and 59 appear an odd number of times, so √649 is irrational and 25.4754784057 is a rounded value.

Where √649 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √649 lies between 25 and 26. 649 is 24 above 625 and 27 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.4706 (0.02% low)
• Tangent from 25, i.e. 25 + 24 ÷ 50: 25.4800 (0.02% high)
• Tangent from 26, i.e. 26 − 27 ÷ 52: 25.4808 (0.02% high)

For √649 the tangent at 25 wins, missing by only 0.0045. Tangent estimates shine when the number sits close to a perfect square — here 649 is just 24 above 625.

Finding √649 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 649: following the tangent line down to zero simplifies to averaging x with 649 ÷ x.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √649 = 25.4754784057 to every decimal shown.

√649 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √649 the pattern is [25; 2, 9, 1, 2, 3, 1, 1, 2, 1, 4, 1, 16, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √649 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 − 649y² = 1. Its smallest solution in positive whole numbers is x = 1,123,593,226,162,199, y = 44,104,892,095,380 — 16 digits for x, even though 649 is small, which is what makes Pell’s equation famous.

√649 in geometry and everyday measurements

• A square garage floor of 649 square feet measures about 25.48 ft (25 ft 6 in) per side, and its corner-to-corner diagonal is √1298 ≈ 36 ft.
• 649 is not a sum of two whole-number squares — the prime factor 11 and 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √649 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 18 × 18 box, because 1² + 18² + 18² = 649.

Square roots near √649 and related roots

• The cube root of 649 is about 8.657947.
• Squaring undoes the root: (√649)² = 649, while 649² = 421,201 — the number whose square root is 649.

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