Square Root of 616

√616 at a glance

Exact value

2√154

Decimal (10 places)

24.8193472920

Rounded

24.8 · 24.82 · 24.819

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.819347

Prime factorization

2³ × 7 × 11

Cube root

8.508642

How to simplify √616

Look for the largest perfect square that divides 616. Here it is 4 (2²), because 616 = 4 × 154 and 154 has no square factor left:

The prime factorization tells the same story: 616 = 2³ × 7 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 7 × 11 stays inside.

Check: (2√154)² = 2² × 154 = 4 × 154 = 616. As a decimal, 2√154 = 2 × 12.409673646 ≈ 24.8193472920.

Where √616 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √616 lies between 24 and 25. 616 is 40 above 576 and 9 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.8163 (0.01% low)
• Tangent from 24, i.e. 24 + 40 ÷ 48: 24.8333 (0.06% high)
• Tangent from 25, i.e. 25 − 9 ÷ 50: 24.8200 (0% high)

For √616 the tangent at 25 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 616 is just 9 below 625.

Finding √616 with the Babylonian method

Picture a rectangle with an area of 616 and one side x; the other side must be 616 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √616.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √616 = 24.8193472920 to every decimal shown.

√616 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √616 the pattern is [24; 1, 4, 1, 1, 6, 1, 1, 4, 1, 48] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √616 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 − 616y² = 1. Its smallest solution in positive whole numbers is x = 21,295, y = 858.

√616 in geometry and everyday measurements

• A square garage floor of 616 square feet measures about 24.82 ft (24 ft 10 in) per side, and its corner-to-corner diagonal is √1232 ≈ 35.1 ft.
• 616 is not a sum of two whole-number squares — the prime factor 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √616 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 24 box, because 2² + 6² + 24² = 616.
• Since √616 = 2√154, a length of √616 is exactly 2 copies of the length √154 laid end to end.

Square roots near √616 and related roots

• The cube root of 616 is about 8.508642.
• Because 616 = 4 × 154, the root is twice √154: 2 × 12.409674 ≈ 24.819347.

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