Square Root of 885

√885 at a glance

Exact value

√885

Decimal (10 places)

29.7489495613

Rounded

29.7 · 29.75 · 29.749

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.748950

Prime factorization

3 × 5 × 59

Cube root

9.600955

How to simplify √885

The prime factorization of 885 is 3 × 5 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √885 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 885, 3, 5 and 59 appear an odd number of times, so √885 is irrational and 29.7489495613 is a rounded value.

Where √885 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √885 lies between 29 and 30. 885 is 44 above 841 and 15 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.7458 (0.01% low)
• Tangent from 29, i.e. 29 + 44 ÷ 58: 29.7586 (0.03% high)
• Tangent from 30, i.e. 30 − 15 ÷ 60: 29.7500 (0% high)

For √885 the tangent at 30 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 885 is just 15 below 900.

Finding √885 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 885: following the tangent line down to zero simplifies to averaging x with 885 ÷ 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 √885 = 29.7489495613 to every decimal shown.

√885 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √885 the pattern is [29; 1, 2, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √885 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 − 885y² = 1. Its smallest solution in positive whole numbers is x = 119, y = 4.

√885 in geometry and everyday measurements

• 885 square feet is 82.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.75 ft (29 ft 9 in) on a side.
• 885 is not a sum of two whole-number squares — the prime factor 3 and 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √885 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 28 box, because 1² + 10² + 28² = 885.

Square roots near √885 and related roots

• The cube root of 885 is about 9.600955.
• Squaring undoes the root: (√885)² = 885, while 885² = 783,225 — the number whose square root is 885.

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