Square Root of 856

√856 at a glance

Exact value

2√214

Decimal (10 places)

29.2574776767

Rounded

29.3 · 29.26 · 29.257

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.257478

Prime factorization

2³ × 107

Cube root

9.494919

How to simplify √856

Look for the largest perfect square that divides 856. Here it is 4 (2²), because 856 = 4 × 214 and 214 has no square factor left:

The prime factorization tells the same story: 856 = 2³ × 107. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 107 stays inside.

Check: (2√214)² = 2² × 214 = 4 × 214 = 856. As a decimal, 2√214 = 2 × 14.6287388383 ≈ 29.2574776767.

Where √856 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √856 lies between 29 and 30. 856 is 15 above 841 and 44 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.2542 (0.01% low)
• Tangent from 29, i.e. 29 + 15 ÷ 58: 29.2586 (0% high)
• Tangent from 30, i.e. 30 − 44 ÷ 60: 29.2667 (0.03% high)

For √856 the tangent at 29 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 856 is just 15 above 841.

Finding √856 with the Babylonian method

Picture a rectangle with an area of 856 and one side x; the other side must be 856 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √856.

Start from the nearest whole number, 29 (29² = 841):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √856 = 29.2574776767 to every decimal shown.

√856 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √856 the pattern is [29; 3, 1, 7, 1, 1, 1, 1, 4, 3, 1, 2, 6, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √856 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 − 856y² = 1. Its smallest solution in positive whole numbers is x = 695,359,189,925, y = 23,766,887,823.

√856 in geometry and everyday measurements

• 856 square feet is 79.5 m². Laid out as a square — a small house footprint or a lot — it is about 29.26 ft (29 ft 3 in) on a side.
• 856 is not a sum of two whole-number squares — the prime factor 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √856 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 6 × 28 box, because 6² + 6² + 28² = 856.
• Since √856 = 2√214, a length of √856 is exactly 2 copies of the length √214 laid end to end.

Square roots near √856 and related roots

• The cube root of 856 is about 9.494919.
• Because 856 = 4 × 214, the root is twice √214: 2 × 14.628739 ≈ 29.257478.

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