Square Root of 844

√844 at a glance

Exact value

2√211

Decimal (10 places)

29.0516780927

Rounded

29.1 · 29.05 · 29.052

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.051678

Prime factorization

2² × 211

Cube root

9.450341

How to simplify √844

Look for the largest perfect square that divides 844. Here it is 4 (2²), because 844 = 4 × 211 and 211 has no square factor left:

The prime factorization tells the same story: 844 = 2² × 211. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 211 stays inside.

Check: (2√211)² = 2² × 211 = 4 × 211 = 844. As a decimal, 2√211 = 2 × 14.5258390463 ≈ 29.0516780927.

Where √844 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √844 lies between 29 and 30. 844 is 3 above 841 and 56 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.0508 (0% low)
• Tangent from 29, i.e. 29 + 3 ÷ 58: 29.0517 (0% high)
• Tangent from 30, i.e. 30 − 56 ÷ 60: 29.0667 (0.05% high)

For √844 the tangent at 29 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 844 is just 3 above 841.

Finding √844 with the Babylonian method

Picture a rectangle with an area of 844 and one side x; the other side must be 844 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √844.

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

Because the starting guess was already close, two steps are enough to match √844 = 29.0516780927 to every decimal shown.

√844 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √844 the pattern is [29; 19, 2, 1, 5, 1, 3, 1, 1, 1, 1, 1, 2, …] with the block of 48 terms after the semicolon repeating forever (only the first 12 of the 48 are shown). A pattern that never ends is one more proof that √844 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 − 844y² = 1. Its smallest solution in positive whole numbers is x = 154,962,314,660,167,628,644,999, y = 5,334,022,845,973,817,148,450 — 24 digits for x, even though 844 is small, which is what makes Pell’s equation famous.

√844 in geometry and everyday measurements

• 844 square feet is 78.4 m². Laid out as a square — a small house footprint or a lot — it is about 29.05 ft (29 ft 1 in) on a side.
• 844 is not a sum of two whole-number squares — the prime factor 211 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √844 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 18 × 22 box, because 6² + 18² + 22² = 844.
• Since √844 = 2√211, a length of √844 is exactly 2 copies of the length √211 laid end to end.

Square roots near √844 and related roots

• The cube root of 844 is about 9.450341.
• Because 844 = 4 × 211, the root is twice √211: 2 × 14.525839 ≈ 29.051678.

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