Square Root of 896

√896 at a glance

Exact value

8√14

Decimal (10 places)

29.9332590942

Rounded

29.9 · 29.93 · 29.933

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.933259

Prime factorization

2⁷ × 7

Cube root

9.640569

How to simplify √896

Look for the largest perfect square that divides 896. Here it is 64 (8²), because 896 = 64 × 14 and 14 has no square factor left:

The prime factorization tells the same story: 896 = 2⁷ × 7. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 2 × 7 stays inside.

896 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √896 = 2√224, and √224 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√14)² = 8² × 14 = 64 × 14 = 896. As a decimal, 8√14 = 8 × 3.7416573868 ≈ 29.9332590942.

Where √896 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √896 lies between 29 and 30. 896 is 55 above 841 and 4 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.9322 (0% low)
• Tangent from 29, i.e. 29 + 55 ÷ 58: 29.9483 (0.05% high)
• Tangent from 30, i.e. 30 − 4 ÷ 60: 29.9333 (0% high)

For √896 the tangent at 30 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 896 is just 4 below 900.

Finding √896 with the Babylonian method

Picture a rectangle with an area of 896 and one side x; the other side must be 896 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √896.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √896 = 29.9332590942 to every decimal shown.

√896 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √896 the pattern is [29; 1, 13, 1, 58] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √896 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 − 896y² = 1. Its smallest solution in positive whole numbers is x = 449, y = 15.

√896 in geometry and everyday measurements

• 896 square feet is 83.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.93 ft (29 ft 11 in) on a side.
• 896 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √896 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 16 × 24 box, because 8² + 16² + 24² = 896.
• Since √896 = 8√14, a length of √896 is exactly 8 copies of the length √14 laid end to end.

Square roots near √896 and related roots

• The cube root of 896 is about 9.640569.
• Because 896 = 4 × 224, the root is twice √224: 2 × 14.96663 ≈ 29.933259.

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