Square Root of 891

√891 at a glance

Exact value

9√11

Decimal (10 places)

29.8496231132

Rounded

29.8 · 29.85 · 29.850

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.849623

Prime factorization

3⁴ × 11

Cube root

9.622603

How to simplify √891

Look for the largest perfect square that divides 891. Here it is 81 (9²), because 891 = 81 × 11 and 11 has no square factor left:

The prime factorization tells the same story: 891 = 3⁴ × 11. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 11 stays inside.

891 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √891 = 3√99, and √99 can be simplified again. Using 81 straight away finishes in one step.

Check: (9√11)² = 9² × 11 = 81 × 11 = 891. As a decimal, 9√11 = 9 × 3.3166247904 ≈ 29.8496231132.

Where √891 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √891 lies between 29 and 30. 891 is 50 above 841 and 9 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.8475 (0.01% low)
• Tangent from 29, i.e. 29 + 50 ÷ 58: 29.8621 (0.04% high)
• Tangent from 30, i.e. 30 − 9 ÷ 60: 29.8500 (0% high)

For √891 the tangent at 30 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 891 is just 9 below 900.

Finding √891 with the Babylonian method

If a guess is too big, 891 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√891) in one step.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √891 = 29.8496231132 to every decimal shown.

√891 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √891 the pattern is [29; 1, 5, 1, 1, 1, 5, 1, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √891 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 − 891y² = 1. Its smallest solution in positive whole numbers is x = 3,970, y = 133.

√891 in geometry and everyday measurements

• 891 square feet is 82.8 m². Laid out as a square — a small house footprint or a lot — it is about 29.85 ft (29 ft 10 in) on a side.
• 891 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 √891 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 29 box, because 1² + 7² + 29² = 891.
• Since √891 = 9√11, a length of √891 is exactly 9 copies of the length √11 laid end to end.

Square roots near √891 and related roots

• The cube root of 891 is about 9.622603.
• Squaring undoes the root: (√891)² = 891, while 891² = 793,881 — the number whose square root is 891.

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