Square Root of 968

√968 at a glance

Exact value

22√2

Decimal (10 places)

31.1126983722

Rounded

31.1 · 31.11 · 31.113

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.112698

Prime factorization

2³ × 11²

Cube root

9.892175

How to simplify √968

Look for the largest perfect square that divides 968. Here it is 484 (22²), because 968 = 484 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 968 = 2³ × 11². Each pair of equal primes leaves the radical as one factor, so 2 × 11 comes out and 2 stays inside.

968 has 3 square factors (4, 121 and 484). Starting with a smaller one still works but takes more rounds: √968 = 2√242, and √242 can be simplified again. Using 484 straight away finishes in one step.

Check: (22√2)² = 22² × 2 = 484 × 2 = 968. As a decimal, 22√2 = 22 × 1.4142135624 ≈ 31.1126983722.

Where √968 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √968 lies between 31 and 32. 968 is 7 above 961 and 56 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.1111 (0.01% low)
• Tangent from 31, i.e. 31 + 7 ÷ 62: 31.1129 (0% high)
• Tangent from 32, i.e. 32 − 56 ÷ 64: 31.1250 (0.04% high)

For √968 the tangent at 31 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 968 is just 7 above 961.

Finding √968 with the Babylonian method

Picture a rectangle with an area of 968 and one side x; the other side must be 968 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √968.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √968 = 31.1126983722 to every decimal shown.

√968 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √968 the pattern is [31; 8, 1, 6, 1, 8, 62] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √968 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 − 968y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 630.

√968 in geometry and everyday measurements

• 968 square feet is 89.9 m². Laid out as a square — a small house footprint or a lot — it is about 31.11 ft (31 ft 1 in) on a side.
• 968 = 22² + 22², so by the Pythagorean theorem √968 is the diagonal of a 22 × 22 rectangle — and the distance between the points (0, 0) and (22, 22) on a grid.
• Since √968 = 22√2, a length of √968 is exactly 22 copies of the length √2 laid end to end.

Square roots near √968 and related roots

• The cube root of 968 is about 9.892175.
• Because 968 = 4 × 242, the root is twice √242: 2 × 15.556349 ≈ 31.112698.

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