Square Root of 640

√640 at a glance

Exact value

8√10

Decimal (10 places)

25.2982212813

Rounded

25.3 · 25.30 · 25.298

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.298221

Prime factorization

2⁷ × 5

Cube root

8.617739

How to simplify √640

Look for the largest perfect square that divides 640. Here it is 64 (8²), because 640 = 64 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 640 = 2⁷ × 5. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 2 × 5 stays inside.

640 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √640 = 2√160, and √160 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√10)² = 8² × 10 = 64 × 10 = 640. As a decimal, 8√10 = 8 × 3.1622776602 ≈ 25.2982212813.

Where √640 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √640 lies between 25 and 26. 640 is 15 above 625 and 36 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.2941 (0.02% low)
• Tangent from 25, i.e. 25 + 15 ÷ 50: 25.3000 (0.01% high)
• Tangent from 26, i.e. 26 − 36 ÷ 52: 25.3077 (0.04% high)

For √640 the tangent at 25 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 640 is just 15 above 625.

Finding √640 with the Babylonian method

Picture a rectangle with an area of 640 and one side x; the other side must be 640 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √640.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √640 = 25.2982212813 to every decimal shown.

√640 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √640 the pattern is [25; 3, 2, 1, 4, 1, 11, 1, 4, 1, 2, 3, 50] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √640 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 − 640y² = 1. Its smallest solution in positive whole numbers is x = 1,039,681, y = 41,097.

√640 in geometry and everyday measurements

• A square garage floor of 640 square feet measures about 25.3 ft (25 ft 4 in) per side, and its corner-to-corner diagonal is √1280 ≈ 35.8 ft.
• 640 = 8² + 24², so by the Pythagorean theorem √640 is the diagonal of a 8 × 24 rectangle — and the distance between the points (0, 0) and (8, 24) on a grid.
• Since √640 = 8√10, a length of √640 is exactly 8 copies of the length √10 laid end to end.

Square roots near √640 and related roots

• The cube root of 640 is about 8.617739.
• Because 640 = 4 × 160, the root is twice √160: 2 × 12.649111 ≈ 25.298221.

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