Square Root of 608

√608 at a glance

Exact value

4√38

Decimal (10 places)

24.6576560119

Rounded

24.7 · 24.66 · 24.658

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.657656

Prime factorization

2⁵ × 19

Cube root

8.471647

How to simplify √608

Look for the largest perfect square that divides 608. Here it is 16 (4²), because 608 = 16 × 38 and 38 has no square factor left:

The prime factorization tells the same story: 608 = 2⁵ × 19. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 19 stays inside.

608 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √608 = 2√152, and √152 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√38)² = 4² × 38 = 16 × 38 = 608. As a decimal, 4√38 = 4 × 6.164414003 ≈ 24.6576560119.

Where √608 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √608 lies between 24 and 25. 608 is 32 above 576 and 17 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.6531 (0.02% low)
• Tangent from 24, i.e. 24 + 32 ÷ 48: 24.6667 (0.04% high)
• Tangent from 25, i.e. 25 − 17 ÷ 50: 24.6600 (0.01% high)

For √608 the tangent at 25 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 608 is just 17 below 625.

Finding √608 with the Babylonian method

Picture a rectangle with an area of 608 and one side x; the other side must be 608 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √608.

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 √608 = 24.6576560119 to every decimal shown.

√608 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √608 the pattern is [24; 1, 1, 1, 11, 1, 1, 1, 48] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √608 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 − 608y² = 1. Its smallest solution in positive whole numbers is x = 2,737, y = 111.

√608 in geometry and everyday measurements

• A square garage floor of 608 square feet measures about 24.66 ft (24 ft 8 in) per side, and its corner-to-corner diagonal is √1216 ≈ 34.9 ft.
• 608 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √608 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 24 box, because 4² + 4² + 24² = 608.
• Since √608 = 4√38, a length of √608 is exactly 4 copies of the length √38 laid end to end.

Square roots near √608 and related roots

• The cube root of 608 is about 8.471647.
• Because 608 = 4 × 152, the root is twice √152: 2 × 12.328828 ≈ 24.657656.

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