Square Root of 796

√796 at a glance

Exact value

2√199

Decimal (10 places)

28.2134719593

Rounded

28.2 · 28.21 · 28.213

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.213472

Prime factorization

2² × 199

Cube root

9.267680

How to simplify √796

Look for the largest perfect square that divides 796. Here it is 4 (2²), because 796 = 4 × 199 and 199 has no square factor left:

The prime factorization tells the same story: 796 = 2² × 199. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 199 stays inside.

Check: (2√199)² = 2² × 199 = 4 × 199 = 796. As a decimal, 2√199 = 2 × 14.1067359797 ≈ 28.2134719593.

Where √796 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √796 lies between 28 and 29. 796 is 12 above 784 and 45 below 841, so the root is closer to 28.

• Straight line between 784 and 841: 28.2105 (0.01% low)
• Tangent from 28, i.e. 28 + 12 ÷ 56: 28.2143 (0% high)
• Tangent from 29, i.e. 29 − 45 ÷ 58: 28.2241 (0.04% high)

For √796 the tangent at 28 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 796 is just 12 above 784.

Finding √796 with the Babylonian method

Picture a rectangle with an area of 796 and one side x; the other side must be 796 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √796.

Start from the nearest whole number, 28 (28² = 784):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √796 = 28.2134719593 to every decimal shown.

√796 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √796 the pattern is [28; 4, 1, 2, 5, 1, 10, 2, 3, 1, 6, 3, 1, …] with the block of 44 terms after the semicolon repeating forever (only the first 12 of the 44 are shown). A pattern that never ends is one more proof that √796 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 − 796y² = 1. Its smallest solution in positive whole numbers is x = 529,178,298,454,520,220,799, y = 18,756,227,493,635,055,480 — 21 digits for x, even though 796 is small, which is what makes Pell’s equation famous.

√796 in geometry and everyday measurements

• 796 square feet is 74 m². Laid out as a square — a small house footprint or a lot — it is about 28.21 ft (28 ft 3 in) on a side.
• 796 is not a sum of two whole-number squares — the prime factor 199 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √796 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √796 as its space diagonal.
• Since √796 = 2√199, a length of √796 is exactly 2 copies of the length √199 laid end to end.

Square roots near √796 and related roots

• The cube root of 796 is about 9.267680.
• Because 796 = 4 × 199, the root is twice √199: 2 × 14.106736 ≈ 28.213472.

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