Square Root of 464

√464 at a glance

Exact value

4√29

Decimal (10 places)

21.5406592285

Rounded

21.5 · 21.54 · 21.541

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.540659

Prime factorization

2⁴ × 29

Cube root

7.741753

How to simplify √464

Look for the largest perfect square that divides 464. Here it is 16 (4²), because 464 = 16 × 29 and 29 has no square factor left:

The prime factorization tells the same story: 464 = 2⁴ × 29. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 29 stays inside.

464 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √464 = 2√116, and √116 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√29)² = 4² × 29 = 16 × 29 = 464. As a decimal, 4√29 = 4 × 5.3851648071 ≈ 21.5406592285.

Where √464 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √464 lies between 21 and 22. 464 is 23 above 441 and 20 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.5349 (0.03% low)
• Tangent from 21, i.e. 21 + 23 ÷ 42: 21.5476 (0.03% high)
• Tangent from 22, i.e. 22 − 20 ÷ 44: 21.5455 (0.02% high)

For √464 the tangent at 22 wins, missing by only 0.0048. Tangent estimates shine when the number sits close to a perfect square — here 464 is just 20 below 484.

Finding √464 with the Babylonian method

Picture a rectangle with an area of 464 and one side x; the other side must be 464 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √464.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √464 = 21.5406592285 to every decimal shown.

√464 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √464 the pattern is [21; 1, 1, 5, 1, 1, 1, 5, 1, 1, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √464 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 − 464y² = 1. Its smallest solution in positive whole numbers is x = 9,801, y = 455.

√464 in geometry and everyday measurements

• A square garage floor of 464 square feet measures about 21.54 ft (21 ft 6 in) per side, and its corner-to-corner diagonal is √928 ≈ 30.5 ft.
• 464 = 8² + 20², so by the Pythagorean theorem √464 is the diagonal of a 8 × 20 rectangle — and the distance between the points (0, 0) and (8, 20) on a grid.
• Since √464 = 4√29, a length of √464 is exactly 4 copies of the length √29 laid end to end.

Square roots near √464 and related roots

• The cube root of 464 is about 7.741753.
• Because 464 = 4 × 116, the root is twice √116: 2 × 10.77033 ≈ 21.540659.

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