Square Root of 96

√96 at a glance

Exact value

4√6

Decimal (10 places)

9.7979589711

Rounded

9.8 · 9.80 · 9.798

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.797959

Prime factorization

2⁵ × 3

Cube root

4.578857

How to simplify √96

Look for the largest perfect square that divides 96. Here it is 16 (4²), because 96 = 16 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 96 = 2⁵ × 3. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 3 stays inside.

96 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √96 = 2√24, and √24 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√6)² = 4² × 6 = 16 × 6 = 96. As a decimal, 4√6 = 4 × 2.4494897428 ≈ 9.7979589711.

Where √96 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √96 lies between 9 and 10. 96 is 15 above 81 and 4 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.7895 (0.09% low)
• Tangent from 9, i.e. 9 + 15 ÷ 18: 9.8333 (0.36% high)
• Tangent from 10, i.e. 10 − 4 ÷ 20: 9.8000 (0.02% high)

For √96 the tangent at 10 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 96 is just 4 below 100.

Finding √96 with the Babylonian method

Picture a rectangle with an area of 96 and one side x; the other side must be 96 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √96.

Start from the nearest whole number, 10 (10² = 100):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √96 = 9.7979589711 to every decimal shown.

√96 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √96 the pattern is [9; 1, 3, 1, 18] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √96 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 − 96y² = 1. Its smallest solution in positive whole numbers is x = 49, y = 5.

√96 in geometry and everyday measurements

• A square room or garden bed covering 96 square feet measures about 9.8 ft (9 ft 10 in) along each wall.
• 96 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √96 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 8 box, because 4² + 4² + 8² = 96.
• Since √96 = 4√6, a length of √96 is exactly 4 copies of the length √6 laid end to end.

Square roots near √96 and related roots

• The cube root of 96 is about 4.578857.
• Four times the radicand doubles the root: √384 = 2 × √96 ≈ 19.595918.

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