Square Root of 560

√560 at a glance

Exact value

4√35

Decimal (10 places)

23.6643191324

Rounded

23.7 · 23.66 · 23.664

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.664319

Prime factorization

2⁴ × 5 × 7

Cube root

8.242571

How to simplify √560

Look for the largest perfect square that divides 560. Here it is 16 (4²), because 560 = 16 × 35 and 35 has no square factor left:

The prime factorization tells the same story: 560 = 2⁴ × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 5 × 7 stays inside.

560 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √560 = 2√140, and √140 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√35)² = 4² × 35 = 16 × 35 = 560. As a decimal, 4√35 = 4 × 5.9160797831 ≈ 23.6643191324.

Where √560 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √560 lies between 23 and 24. 560 is 31 above 529 and 16 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.6596 (0.02% low)
• Tangent from 23, i.e. 23 + 31 ÷ 46: 23.6739 (0.04% high)
• Tangent from 24, i.e. 24 − 16 ÷ 48: 23.6667 (0.01% high)

For √560 the tangent at 24 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 560 is just 16 below 576.

Finding √560 with the Babylonian method

Picture a rectangle with an area of 560 and one side x; the other side must be 560 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √560.

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 √560 = 23.6643191324 to every decimal shown.

√560 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √560 the pattern is [23; 1, 1, 1, 46] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √560 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 − 560y² = 1. Its smallest solution in positive whole numbers is x = 71, y = 3.

√560 in geometry and everyday measurements

• A square garage floor of 560 square feet measures about 23.66 ft (23 ft 8 in) per side, and its corner-to-corner diagonal is √1120 ≈ 33.5 ft.
• 560 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √560 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 12 × 20 box, because 4² + 12² + 20² = 560.
• Since √560 = 4√35, a length of √560 is exactly 4 copies of the length √35 laid end to end.

Square roots near √560 and related roots

• The cube root of 560 is about 8.242571.
• Because 560 = 4 × 140, the root is twice √140: 2 × 11.83216 ≈ 23.664319.

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