Square Root of 833

√833 at a glance

Exact value

7√17

Decimal (10 places)

28.8617393793

Rounded

28.9 · 28.86 · 28.862

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.861739

Prime factorization

7² × 17

Cube root

9.409105

How to simplify √833

Look for the largest perfect square that divides 833. Here it is 49 (7²), because 833 = 49 × 17 and 17 has no square factor left:

The prime factorization tells the same story: 833 = 7² × 17. Each pair of equal primes leaves the radical as one factor, so 7 comes out and 17 stays inside.

Check: (7√17)² = 7² × 17 = 49 × 17 = 833. As a decimal, 7√17 = 7 × 4.1231056256 ≈ 28.8617393793.

Where √833 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √833 lies between 28 and 29. 833 is 49 above 784 and 8 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.8596 (0.01% low)
• Tangent from 28, i.e. 28 + 49 ÷ 56: 28.8750 (0.05% high)
• Tangent from 29, i.e. 29 − 8 ÷ 58: 28.8621 (0% high)

For √833 the tangent at 29 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 833 is just 8 below 841.

Finding √833 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 833: following the tangent line down to zero simplifies to averaging x with 833 ÷ x.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √833 = 28.8617393793 to every decimal shown.

√833 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √833 the pattern is [28; 1, 6, 4, 3, 2, 1, 2, 1, 2, 3, 4, 6, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √833 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 − 833y² = 1. Its smallest solution in positive whole numbers is x = 9,478,657, y = 328,416.

√833 in geometry and everyday measurements

• 833 square feet is 77.4 m². Laid out as a square — a small house footprint or a lot — it is about 28.86 ft (28 ft 10 in) on a side.
• 833 = 7² + 28², so by the Pythagorean theorem √833 is the diagonal of a 7 × 28 rectangle — and the distance between the points (0, 0) and (7, 28) on a grid.
• Since √833 = 7√17, a length of √833 is exactly 7 copies of the length √17 laid end to end.

Square roots near √833 and related roots

• The cube root of 833 is about 9.409105.
• Squaring undoes the root: (√833)² = 833, while 833² = 693,889 — the number whose square root is 833.

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