Square Root of 836

√836 at a glance

Exact value

2√209

Decimal (10 places)

28.9136645896

Rounded

28.9 · 28.91 · 28.914

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.913665

Prime factorization

2² × 11 × 19

Cube root

9.420387

How to simplify √836

Look for the largest perfect square that divides 836. Here it is 4 (2²), because 836 = 4 × 209 and 209 has no square factor left:

The prime factorization tells the same story: 836 = 2² × 11 × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 11 × 19 stays inside.

Check: (2√209)² = 2² × 209 = 4 × 209 = 836. As a decimal, 2√209 = 2 × 14.4568322948 ≈ 28.9136645896.

Where √836 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √836 lies between 28 and 29. 836 is 52 above 784 and 5 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.9123 (0% low)
• Tangent from 28, i.e. 28 + 52 ÷ 56: 28.9286 (0.05% high)
• Tangent from 29, i.e. 29 − 5 ÷ 58: 28.9138 (0% high)

For √836 the tangent at 29 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 836 is just 5 below 841.

Finding √836 with the Babylonian method

Picture a rectangle with an area of 836 and one side x; the other side must be 836 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √836.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √836 = 28.9136645896 to every decimal shown.

√836 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √836 the pattern is [28; 1, 10, 1, 1, 2, 1, 1, 10, 1, 56] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √836 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 − 836y² = 1. Its smallest solution in positive whole numbers is x = 46,551, y = 1,610.

√836 in geometry and everyday measurements

• 836 square feet is 77.7 m². Laid out as a square — a small house footprint or a lot — it is about 28.91 ft (28 ft 11 in) on a side.
• 836 is not a sum of two whole-number squares — the prime factor 11 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √836 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 16 × 24 box, because 2² + 16² + 24² = 836.
• Since √836 = 2√209, a length of √836 is exactly 2 copies of the length √209 laid end to end.

Square roots near √836 and related roots

• The cube root of 836 is about 9.420387.
• Because 836 = 4 × 209, the root is twice √209: 2 × 14.456832 ≈ 28.913665.

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