Square Root of 99

√99 at a glance

Exact value

3√11

Decimal (10 places)

9.9498743711

Rounded

9.9 · 9.95 · 9.950

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.949874

Prime factorization

3² × 11

Cube root

4.626065

How to simplify √99

Look for the largest perfect square that divides 99. Here it is 9 (3²), because 99 = 9 × 11 and 11 has no square factor left:

The prime factorization tells the same story: 99 = 3² × 11. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 11 stays inside.

Check: (3√11)² = 3² × 11 = 9 × 11 = 99. As a decimal, 3√11 = 3 × 3.3166247904 ≈ 9.9498743711.

Where √99 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √99 lies between 9 and 10. 99 is 18 above 81 and 1 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.9474 (0.03% low)
• Tangent from 9, i.e. 9 + 18 ÷ 18: 10.0000 (0.5% high)
• Tangent from 10, i.e. 10 − 1 ÷ 20: 9.9500 (0% high)

For √99 the tangent at 10 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 99 is just 1 below 100.

Finding √99 with the Babylonian method

If a guess is too big, 99 ÷ 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√99) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √99 = 9.9498743711 to every decimal shown.

√99 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √99 the pattern is [9; 1, 18] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √99 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 − 99y² = 1. Its smallest solution in positive whole numbers is x = 10, y = 1.

√99 in geometry and everyday measurements

• A square room or garden bed covering 99 square feet measures about 9.95 ft (9 ft 11 in) along each wall.
• 99 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √99 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 7 box, because 1² + 7² + 7² = 99.
• Since √99 = 3√11, a length of √99 is exactly 3 copies of the length √11 laid end to end.

Square roots near √99 and related roots

• The cube root of 99 is about 4.626065.
• Four times the radicand doubles the root: √396 = 2 × √99 ≈ 19.899749.

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