Square Root of 870

√870 at a glance

Exact value

√870

Decimal (10 places)

29.4957624075

Rounded

29.5 · 29.50 · 29.496

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.495762

Prime factorization

2 × 3 × 5 × 29

Cube root

9.546403

How to simplify √870

The prime factorization of 870 is 2 × 3 × 5 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √870 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 870, 2, 3, 5 and 29 appear an odd number of times, so √870 is irrational and 29.4957624075 is a rounded value.

Where √870 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √870 lies between 29 and 30. 870 is 29 above 841 and 30 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.4915 (0.01% low)
• Tangent from 29, i.e. 29 + 29 ÷ 58: 29.5000 (0.01% high)
• Tangent from 30, i.e. 30 − 30 ÷ 60: 29.5000 (0.01% high)

For √870 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √870 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √870 = 29.4957624075 to every decimal shown.

√870 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √870 the pattern is [29; 2, 58] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √870 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 − 870y² = 1. Its smallest solution in positive whole numbers is x = 59, y = 2.

√870 in geometry and everyday measurements

• 870 square feet is 80.8 m². Laid out as a square — a small house footprint or a lot — it is about 29.5 ft (29 ft 6 in) on a side.
• 870 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √870 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 29 box, because 2² + 5² + 29² = 870.

Square roots near √870 and related roots

• The cube root of 870 is about 9.546403.
• Squaring undoes the root: (√870)² = 870, while 870² = 756,900 — the number whose square root is 870.

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