Square Root of 986

√986 at a glance

Exact value

√986

Decimal (10 places)

31.4006369362

Rounded

31.4 · 31.40 · 31.401

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.400637

Prime factorization

2 × 17 × 29

Cube root

9.953114

How to simplify √986

The prime factorization of 986 is 2 × 17 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √986 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 986, 2, 17 and 29 appear an odd number of times, so √986 is irrational and 31.4006369362 is a rounded value.

Where √986 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √986 lies between 31 and 32. 986 is 25 above 961 and 38 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.3968 (0.01% low)
• Tangent from 31, i.e. 31 + 25 ÷ 62: 31.4032 (0.01% high)
• Tangent from 32, i.e. 32 − 38 ÷ 64: 31.4063 (0.02% high)

For √986 the tangent at 31 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 986 is just 25 above 961.

Finding √986 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, 31 (31² = 961):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √986 = 31.4006369362 to every decimal shown.

√986 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √986 the pattern is [31; 2, 2, 62] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √986 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 − 986y² = 1. Its smallest solution in positive whole numbers is x = 49,299, y = 1,570. Because the period is odd, the equation with −1 on the right also has a solution: 157² − 986 × 5² = −1.

√986 in geometry and everyday measurements

• 986 square feet is 91.6 m². Laid out as a square — a small house footprint or a lot — it is about 31.4 ft (31 ft 5 in) on a side.
• 986 = 5² + 31² = 19² + 25², so by the Pythagorean theorem √986 is the diagonal of rectangles measuring 5 × 31 and 19 × 25 — and the distance between the points (0, 0) and (5, 31) on a grid.

Square roots near √986 and related roots

• The cube root of 986 is about 9.953114.
• Squaring undoes the root: (√986)² = 986, while 986² = 972,196 — the number whose square root is 986.

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