Square Root of 842

√842 at a glance

Exact value

√842

Decimal (10 places)

29.0172362571

Rounded

29.0 · 29.02 · 29.017

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.017236

Prime factorization

2 × 421

Cube root

9.442870

How to simplify √842

The prime factorization of 842 is 2 × 421. Every prime appears only once, so there is no pair to bring outside the radical — √842 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 842, 2 and 421 appear an odd number of times, so √842 is irrational and 29.0172362571 is a rounded value.

Where √842 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √842 lies between 29 and 30. 842 is 1 above 841 and 58 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.0169 (0% low)
• Tangent from 29, i.e. 29 + 1 ÷ 58: 29.0172 (0% high)
• Tangent from 30, i.e. 30 − 58 ÷ 60: 29.0333 (0.06% high)

For √842 the tangent at 29 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 842 is just 1 above 841.

Finding √842 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):

Because the starting guess was already close, two steps are enough to match √842 = 29.0172362571 to every decimal shown.

√842 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √842 the pattern is [29; 58] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 842 is one more than a perfect square (29² + 1). A pattern that never ends is one more proof that √842 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 − 842y² = 1. Its smallest solution in positive whole numbers is x = 1,683, y = 58. Because the period is odd, the equation with −1 on the right also has a solution: 29² − 842 × 1² = −1.

√842 in geometry and everyday measurements

• 842 square feet is 78.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.02 ft (29 ft) on a side.
• 842 = 1² + 29², so by the Pythagorean theorem √842 is the diagonal of a 1 × 29 rectangle — and the distance between the points (0, 0) and (1, 29) on a grid.

Square roots near √842 and related roots

• The cube root of 842 is about 9.442870.
• Squaring undoes the root: (√842)² = 842, while 842² = 708,964 — the number whose square root is 842.

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