Square Root of 792

√792 at a glance

Exact value

6√22

Decimal (10 places)

28.1424945589

Rounded

28.1 · 28.14 · 28.142

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.142495

Prime factorization

2³ × 3² × 11

Cube root

9.252130

How to simplify √792

Look for the largest perfect square that divides 792. Here it is 36 (6²), because 792 = 36 × 22 and 22 has no square factor left:

The prime factorization tells the same story: 792 = 2³ × 3² × 11. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 2 × 11 stays inside.

792 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √792 = 2√198, and √198 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√22)² = 6² × 22 = 36 × 22 = 792. As a decimal, 6√22 = 6 × 4.6904157598 ≈ 28.1424945589.

Where √792 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √792 lies between 28 and 29. 792 is 8 above 784 and 49 below 841, so the root is closer to 28.

• Straight line between 784 and 841: 28.1404 (0.01% low)
• Tangent from 28, i.e. 28 + 8 ÷ 56: 28.1429 (0% high)
• Tangent from 29, i.e. 29 − 49 ÷ 58: 28.1552 (0.05% high)

For √792 the tangent at 28 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 792 is just 8 above 784.

Finding √792 with the Babylonian method

Picture a rectangle with an area of 792 and one side x; the other side must be 792 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √792.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √792 = 28.1424945589 to every decimal shown.

√792 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √792 the pattern is [28; 7, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √792 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 − 792y² = 1. Its smallest solution in positive whole numbers is x = 197, y = 7.

√792 in geometry and everyday measurements

• 792 square feet is 73.6 m². Laid out as a square — a small house footprint or a lot — it is about 28.14 ft (28 ft 2 in) on a side.
• 792 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √792 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 28 box, because 2² + 2² + 28² = 792.
• Since √792 = 6√22, a length of √792 is exactly 6 copies of the length √22 laid end to end.

Square roots near √792 and related roots

• The cube root of 792 is about 9.252130.
• Because 792 = 4 × 198, the root is twice √198: 2 × 14.071247 ≈ 28.142495.

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