Square Root of 603

√603 at a glance

Exact value

3√67

Decimal (10 places)

24.5560583156

Rounded

24.6 · 24.56 · 24.556

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.556058

Prime factorization

3² × 67

Cube root

8.448361

How to simplify √603

Look for the largest perfect square that divides 603. Here it is 9 (3²), because 603 = 9 × 67 and 67 has no square factor left:

The prime factorization tells the same story: 603 = 3² × 67. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 67 stays inside.

Check: (3√67)² = 3² × 67 = 9 × 67 = 603. As a decimal, 3√67 = 3 × 8.1853527719 ≈ 24.5560583156.

Where √603 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √603 lies between 24 and 25. 603 is 27 above 576 and 22 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.5510 (0.02% low)
• Tangent from 24, i.e. 24 + 27 ÷ 48: 24.5625 (0.03% high)
• Tangent from 25, i.e. 25 − 22 ÷ 50: 24.5600 (0.02% high)

For √603 the tangent at 25 wins, missing by only 0.0039. Tangent estimates shine when the number sits close to a perfect square — here 603 is just 22 below 625.

Finding √603 with the Babylonian method

If a guess is too big, 603 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√603) in one step.

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 √603 = 24.5560583156 to every decimal shown.

√603 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √603 the pattern is [24; 1, 1, 3, 1, 23, 1, 3, 1, 1, 48] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √603 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 − 603y² = 1. Its smallest solution in positive whole numbers is x = 48,842, y = 1,989.

√603 in geometry and everyday measurements

• A square garage floor of 603 square feet measures about 24.56 ft (24 ft 7 in) per side, and its corner-to-corner diagonal is √1206 ≈ 34.7 ft.
• 603 is not a sum of two whole-number squares — the prime factor 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √603 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 7 × 23 box, because 5² + 7² + 23² = 603.
• Since √603 = 3√67, a length of √603 is exactly 3 copies of the length √67 laid end to end.

Square roots near √603 and related roots

• The cube root of 603 is about 8.448361.
• Squaring undoes the root: (√603)² = 603, while 603² = 363,609 — the number whose square root is 603.

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