Square Root of 56

√56 at a glance

Exact value

2√14

Decimal (10 places)

7.4833147735

Rounded

7.5 · 7.48 · 7.483

Perfect square?

No — between 7² and 8²

Rational?

Irrational

Both square roots

±7.483315

Prime factorization

2³ × 7

Cube root

3.825862

How to simplify √56

Look for the largest perfect square that divides 56. Here it is 4 (2²), because 56 = 4 × 14 and 14 has no square factor left:

The prime factorization tells the same story: 56 = 2³ × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 7 stays inside.

Check: (2√14)² = 2² × 14 = 4 × 14 = 56. As a decimal, 2√14 = 2 × 3.7416573868 ≈ 7.4833147735.

Where √56 sits between perfect squares

49 = 7² and 64 = 8² are the nearest perfect squares, so √56 lies between 7 and 8. 56 is 7 above 49 and 8 below 64, so the root is closer to 7.

• Straight line between 49 and 64: 7.4667 (0.22% low)
• Tangent from 7, i.e. 7 + 7 ÷ 14: 7.5000 (0.22% high)
• Tangent from 8, i.e. 8 − 8 ÷ 16: 7.5000 (0.22% high)

For √56 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 √56 with the Babylonian method

Picture a rectangle with an area of 56 and one side x; the other side must be 56 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √56.

Start from the nearest whole number, 7 (7² = 49):

The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √56 = 7.4833147735 to every decimal shown.

√56 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √56 the pattern is [7; 2, 14] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √56 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 − 56y² = 1. Its smallest solution in positive whole numbers is x = 15, y = 2.

√56 in geometry and everyday measurements

• A square room or garden bed covering 56 square feet measures about 7.48 ft (7 ft 6 in) along each wall.
• 56 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 √56 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 6 box, because 2² + 4² + 6² = 56.
• Since √56 = 2√14, a length of √56 is exactly 2 copies of the length √14 laid end to end.

Square roots near √56 and related roots

• The cube root of 56 is about 3.825862.
• Four times the radicand doubles the root: √224 = 2 × √56 ≈ 14.96663.

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