Square Root of 645

√645 at a glance

Exact value

√645

Decimal (10 places)

25.3968501984

Rounded

25.4 · 25.40 · 25.397

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.396850

Prime factorization

3 × 5 × 43

Cube root

8.640123

How to simplify √645

The prime factorization of 645 is 3 × 5 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √645 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 645, 3, 5 and 43 appear an odd number of times, so √645 is irrational and 25.3968501984 is a rounded value.

Where √645 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √645 lies between 25 and 26. 645 is 20 above 625 and 31 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.3922 (0.02% low)
• Tangent from 25, i.e. 25 + 20 ÷ 50: 25.4000 (0.01% high)
• Tangent from 26, i.e. 26 − 31 ÷ 52: 25.4038 (0.03% high)

For √645 the tangent at 25 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 645 is just 20 above 625.

Finding √645 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 645: following the tangent line down to zero simplifies to averaging x with 645 ÷ 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 √645 = 25.3968501984 to every decimal shown.

√645 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √645 the pattern is [25; 2, 1, 1, 12, 10, 12, 1, 1, 2, 50] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √645 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 − 645y² = 1. Its smallest solution in positive whole numbers is x = 1,024,001, y = 40,320.

√645 in geometry and everyday measurements

• A square garage floor of 645 square feet measures about 25.4 ft (25 ft 5 in) per side, and its corner-to-corner diagonal is √1290 ≈ 35.9 ft.
• 645 is not a sum of two whole-number squares — the prime factor 3 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √645 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 25 box, because 2² + 4² + 25² = 645.

Square roots near √645 and related roots

• The cube root of 645 is about 8.640123.
• Squaring undoes the root: (√645)² = 645, while 645² = 416,025 — the number whose square root is 645.

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