Square Root of 837

√837 at a glance

Exact value

3√93

Decimal (10 places)

28.9309522830

Rounded

28.9 · 28.93 · 28.931

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.930952

Prime factorization

3³ × 31

Cube root

9.424142

How to simplify √837

Look for the largest perfect square that divides 837. Here it is 9 (3²), because 837 = 9 × 93 and 93 has no square factor left:

The prime factorization tells the same story: 837 = 3³ × 31. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 31 stays inside.

Check: (3√93)² = 3² × 93 = 9 × 93 = 837. As a decimal, 3√93 = 3 × 9.643650761 ≈ 28.9309522830.

Where √837 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √837 lies between 28 and 29. 837 is 53 above 784 and 4 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.9298 (0% low)
• Tangent from 28, i.e. 28 + 53 ÷ 56: 28.9464 (0.05% high)
• Tangent from 29, i.e. 29 − 4 ÷ 58: 28.9310 (0% high)

For √837 the tangent at 29 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 837 is just 4 below 841.

Finding √837 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 837: following the tangent line down to zero simplifies to averaging x with 837 ÷ x.

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

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √837 = 28.9309522830 to every decimal shown.

√837 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √837 the pattern is [28; 1, 13, 2, 13, 1, 56] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √837 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 − 837y² = 1. Its smallest solution in positive whole numbers is x = 12,151, y = 420.

√837 in geometry and everyday measurements

• 837 square feet is 77.8 m². Laid out as a square — a small house footprint or a lot — it is about 28.93 ft (28 ft 11 in) on a side.
• 837 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √837 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 28 box, because 2² + 7² + 28² = 837.
• Since √837 = 3√93, a length of √837 is exactly 3 copies of the length √93 laid end to end.

Square roots near √837 and related roots

• The cube root of 837 is about 9.424142.
• Squaring undoes the root: (√837)² = 837, while 837² = 700,569 — the number whose square root is 837.

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