Square Root of 881

√881 at a glance

Exact value

√881

Decimal (10 places)

29.6816441593

Rounded

29.7 · 29.68 · 29.682

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.681644

Prime factorization

881

Cube root

9.586468

How to simplify √881

881 is a prime number, so its only factors are 1 and 881. There is no perfect-square factor to pull out, which means √881 is already in its simplest radical form.

The square root of any prime is irrational. If √881 were a fraction a/b in lowest terms, then a² = 881b², so 881 would divide a — and then 881 would divide b too, contradicting “lowest terms.” That is why the decimal 29.6816441593 is only a rounded value.

Where √881 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √881 lies between 29 and 30. 881 is 40 above 841 and 19 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.6780 (0.01% low)
• Tangent from 29, i.e. 29 + 40 ÷ 58: 29.6897 (0.03% high)
• Tangent from 30, i.e. 30 − 19 ÷ 60: 29.6833 (0.01% high)

For √881 the tangent at 30 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 881 is just 19 below 900.

Finding √881 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 881: following the tangent line down to zero simplifies to averaging x with 881 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √881 = 29.6816441593 to every decimal shown.

√881 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √881 the pattern is [29; 1, 2, 7, 11, 1, 2, 1, 3, 1, 4, 1, 1, …] with the block of 25 terms after the semicolon repeating forever (only the first 12 of the 25 are shown). A pattern that never ends is one more proof that √881 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 − 881y² = 1. Its smallest solution in positive whole numbers is x = 22,606,256,615,916,825,861,249, y = 761,624,136,944,072,910,800 — 23 digits for x, even though 881 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 106,316,171,432² − 881 × 3,581,882,825² = −1.

√881 in geometry and everyday measurements

• 881 square feet is 81.8 m². Laid out as a square — a small house footprint or a lot — it is about 29.68 ft (29 ft 8 in) on a side.
• 881 = 16² + 25², so by the Pythagorean theorem √881 is the diagonal of a 16 × 25 rectangle — and the distance between the points (0, 0) and (16, 25) on a grid.

Square roots near √881 and related roots

• The cube root of 881 is about 9.586468.
• Squaring undoes the root: (√881)² = 881, while 881² = 776,161 — the number whose square root is 881.

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