Square Root of 600

√600 at a glance

Exact value

10√6

Decimal (10 places)

24.4948974278

Rounded

24.5 · 24.49 · 24.495

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.494897

Prime factorization

2³ × 3 × 5²

Cube root

8.434327

How to simplify √600

Look for the largest perfect square that divides 600. Here it is 100 (10²), because 600 = 100 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 600 = 2³ × 3 × 5². Each pair of equal primes leaves the radical as one factor, so 2 × 5 comes out and 2 × 3 stays inside.

600 has 3 square factors (4, 25 and 100). Starting with a smaller one still works but takes more rounds: √600 = 2√150, and √150 can be simplified again. Using 100 straight away finishes in one step.

Check: (10√6)² = 10² × 6 = 100 × 6 = 600. As a decimal, 10√6 = 10 × 2.4494897428 ≈ 24.4948974278.

Where √600 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √600 lies between 24 and 25. 600 is 24 above 576 and 25 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.4898 (0.02% low)
• Tangent from 24, i.e. 24 + 24 ÷ 48: 24.5000 (0.02% high)
• Tangent from 25, i.e. 25 − 25 ÷ 50: 24.5000 (0.02% high)

For √600 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √600 with the Babylonian method

Picture a rectangle with an area of 600 and one side x; the other side must be 600 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √600.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √600 = 24.4948974278 to every decimal shown.

√600 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √600 the pattern is [24; 2, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √600 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 − 600y² = 1. Its smallest solution in positive whole numbers is x = 49, y = 2.

√600 in geometry and everyday measurements

• A square garage floor of 600 square feet measures about 24.49 ft (24 ft 6 in) per side, and its corner-to-corner diagonal is √1200 ≈ 34.6 ft.
• 600 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √600 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 14 × 20 box, because 2² + 14² + 20² = 600.
• Since √600 = 10√6, a length of √600 is exactly 10 copies of the length √6 laid end to end.

Square roots near √600 and related roots

• The cube root of 600 is about 8.434327.
• Dividing by 100 divides the root by 10: √6 = √600 ÷ 10 ≈ 2.44948974.

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