Square Root of 828

√828 at a glance

Exact value

6√23

Decimal (10 places)

28.7749891399

Rounded

28.8 · 28.77 · 28.775

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.774989

Prime factorization

2² × 3² × 23

Cube root

9.390242

How to simplify √828

Look for the largest perfect square that divides 828. Here it is 36 (6²), because 828 = 36 × 23 and 23 has no square factor left:

The prime factorization tells the same story: 828 = 2² × 3² × 23. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 23 stays inside.

828 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √828 = 2√207, and √207 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√23)² = 6² × 23 = 36 × 23 = 828. As a decimal, 6√23 = 6 × 4.7958315233 ≈ 28.7749891399.

Where √828 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √828 lies between 28 and 29. 828 is 44 above 784 and 13 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.7719 (0.01% low)
• Tangent from 28, i.e. 28 + 44 ÷ 56: 28.7857 (0.04% high)
• Tangent from 29, i.e. 29 − 13 ÷ 58: 28.7759 (0% high)

For √828 the tangent at 29 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 828 is just 13 below 841.

Finding √828 with the Babylonian method

Picture a rectangle with an area of 828 and one side x; the other side must be 828 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √828.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √828 = 28.7749891399 to every decimal shown.

√828 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √828 the pattern is [28; 1, 3, 2, 3, 1, 56] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √828 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 − 828y² = 1. Its smallest solution in positive whole numbers is x = 1,151, y = 40.

√828 in geometry and everyday measurements

• 828 square feet is 76.9 m². Laid out as a square — a small house footprint or a lot — it is about 28.77 ft (28 ft 9 in) on a side.
• 828 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √828 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 √828 as its space diagonal.
• Since √828 = 6√23, a length of √828 is exactly 6 copies of the length √23 laid end to end.

Square roots near √828 and related roots

• The cube root of 828 is about 9.390242.
• Because 828 = 4 × 207, the root is twice √207: 2 × 14.387495 ≈ 28.774989.

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