Square Root of 981

√981 at a glance

Exact value

3√109

Decimal (10 places)

31.3209195267

Rounded

31.3 · 31.32 · 31.321

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.320920

Prime factorization

3² × 109

Cube root

9.936261

How to simplify √981

Look for the largest perfect square that divides 981. Here it is 9 (3²), because 981 = 9 × 109 and 109 has no square factor left:

The prime factorization tells the same story: 981 = 3² × 109. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 109 stays inside.

Check: (3√109)² = 3² × 109 = 9 × 109 = 981. As a decimal, 3√109 = 3 × 10.4403065089 ≈ 31.3209195267.

Where √981 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √981 lies between 31 and 32. 981 is 20 above 961 and 43 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.3175 (0.01% low)
• Tangent from 31, i.e. 31 + 20 ÷ 62: 31.3226 (0.01% high)
• Tangent from 32, i.e. 32 − 43 ÷ 64: 31.3281 (0.02% high)

For √981 the tangent at 31 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 981 is just 20 above 961.

Finding √981 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 981: following the tangent line down to zero simplifies to averaging x with 981 ÷ x.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √981 = 31.3209195267 to every decimal shown.

√981 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √981 the pattern is [31; 3, 8, 1, 1, 1, 1, 1, 1, 2, 1, 1, 15, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √981 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 − 981y² = 1. Its smallest solution in positive whole numbers is x = 158,070,671,986,249, y = 5,046,808,151,700 — 15 digits for x, even though 981 is small, which is what makes Pell’s equation famous.

√981 in geometry and everyday measurements

• 981 square feet is 91.1 m². Laid out as a square — a small house footprint or a lot — it is about 31.32 ft (31 ft 4 in) on a side.
• 981 = 9² + 30², so by the Pythagorean theorem √981 is the diagonal of a 9 × 30 rectangle — and the distance between the points (0, 0) and (9, 30) on a grid.
• Since √981 = 3√109, a length of √981 is exactly 3 copies of the length √109 laid end to end.

Square roots near √981 and related roots

• The cube root of 981 is about 9.936261.
• Squaring undoes the root: (√981)² = 981, while 981² = 962,361 — the number whose square root is 981.

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