Square Root of 963

√963 at a glance

Exact value

3√107

Decimal (10 places)

31.0322412984

Rounded

31.0 · 31.03 · 31.032

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.032241

Prime factorization

3² × 107

Cube root

9.875113

How to simplify √963

Look for the largest perfect square that divides 963. Here it is 9 (3²), because 963 = 9 × 107 and 107 has no square factor left:

The prime factorization tells the same story: 963 = 3² × 107. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 107 stays inside.

Check: (3√107)² = 3² × 107 = 9 × 107 = 963. As a decimal, 3√107 = 3 × 10.3440804328 ≈ 31.0322412984.

Where √963 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √963 lies between 31 and 32. 963 is 2 above 961 and 61 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.0317 (0% low)
• Tangent from 31, i.e. 31 + 2 ÷ 62: 31.0323 (0% high)
• Tangent from 32, i.e. 32 − 61 ÷ 64: 31.0469 (0.05% high)

For √963 the tangent at 31 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 963 is just 2 above 961.

Finding √963 with the Babylonian method

If a guess is too big, 963 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√963) in one step.

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

Because the starting guess was already close, two steps are enough to match √963 = 31.0322412984 to every decimal shown.

√963 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √963 the pattern is [31; 31, 62] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √963 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 − 963y² = 1. Its smallest solution in positive whole numbers is x = 962, y = 31.

√963 in geometry and everyday measurements

• 963 square feet is 89.5 m². Laid out as a square — a small house footprint or a lot — it is about 31.03 ft (31 ft) on a side.
• 963 is not a sum of two whole-number squares — the prime factor 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √963 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 31 box, because 1² + 1² + 31² = 963.
• Since √963 = 3√107, a length of √963 is exactly 3 copies of the length √107 laid end to end.

Square roots near √963 and related roots

• The cube root of 963 is about 9.875113.
• Squaring undoes the root: (√963)² = 963, while 963² = 927,369 — the number whose square root is 963.

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