Square Root of 873

√873 at a glance

Exact value

3√97

Decimal (10 places)

29.5465734054

Rounded

29.5 · 29.55 · 29.547

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.546573

Prime factorization

3² × 97

Cube root

9.557363

How to simplify √873

Look for the largest perfect square that divides 873. Here it is 9 (3²), because 873 = 9 × 97 and 97 has no square factor left:

The prime factorization tells the same story: 873 = 3² × 97. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 97 stays inside.

Check: (3√97)² = 3² × 97 = 9 × 97 = 873. As a decimal, 3√97 = 3 × 9.8488578018 ≈ 29.5465734054.

Where √873 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √873 lies between 29 and 30. 873 is 32 above 841 and 27 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.5424 (0.01% low)
• Tangent from 29, i.e. 29 + 32 ÷ 58: 29.5517 (0.02% high)
• Tangent from 30, i.e. 30 − 27 ÷ 60: 29.5500 (0.01% high)

For √873 the tangent at 30 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 873 is just 27 below 900.

Finding √873 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 873: following the tangent line down to zero simplifies to averaging x with 873 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √873 = 29.5465734054 to every decimal shown.

√873 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √873 the pattern is [29; 1, 1, 4, 1, 6, 1, 1, 3, 6, 3, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √873 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 − 873y² = 1. Its smallest solution in positive whole numbers is x = 62,809,633, y = 2,125,784.

√873 in geometry and everyday measurements

• 873 square feet is 81.1 m². Laid out as a square — a small house footprint or a lot — it is about 29.55 ft (29 ft 7 in) on a side.
• 873 = 12² + 27², so by the Pythagorean theorem √873 is the diagonal of a 12 × 27 rectangle — and the distance between the points (0, 0) and (12, 27) on a grid.
• Since √873 = 3√97, a length of √873 is exactly 3 copies of the length √97 laid end to end.

Square roots near √873 and related roots

• The cube root of 873 is about 9.557363.
• Squaring undoes the root: (√873)² = 873, while 873² = 762,129 — the number whose square root is 873.

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