Square Root of 602

√602 at a glance

Exact value

√602

Decimal (10 places)

24.5356882928

Rounded

24.5 · 24.54 · 24.536

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.535688

Prime factorization

2 × 7 × 43

Cube root

8.443688

How to simplify √602

The prime factorization of 602 is 2 × 7 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √602 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 602, 2, 7 and 43 appear an odd number of times, so √602 is irrational and 24.5356882928 is a rounded value.

Where √602 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √602 lies between 24 and 25. 602 is 26 above 576 and 23 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.5306 (0.02% low)
• Tangent from 24, i.e. 24 + 26 ÷ 48: 24.5417 (0.02% high)
• Tangent from 25, i.e. 25 − 23 ÷ 50: 24.5400 (0.02% high)

For √602 the tangent at 25 wins, missing by only 0.0043. Tangent estimates shine when the number sits close to a perfect square — here 602 is just 23 below 625.

Finding √602 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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 √602 = 24.5356882928 to every decimal shown.

√602 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √602 the pattern is [24; 1, 1, 6, 1, 1, 48] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √602 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 − 602y² = 1. Its smallest solution in positive whole numbers is x = 687, y = 28.

√602 in geometry and everyday measurements

• A square garage floor of 602 square feet measures about 24.54 ft (24 ft 6 in) per side, and its corner-to-corner diagonal is √1204 ≈ 34.7 ft.
• 602 is not a sum of two whole-number squares — the prime factor 7 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √602 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 24 box, because 1² + 5² + 24² = 602.

Square roots near √602 and related roots

• The cube root of 602 is about 8.443688.
• Squaring undoes the root: (√602)² = 602, while 602² = 362,404 — the number whose square root is 602.

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