Square Root of 93

√93 at a glance

Exact value

√93

Decimal (10 places)

9.6436507610

Rounded

9.6 · 9.64 · 9.644

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.643651

Prime factorization

3 × 31

Cube root

4.530655

How to simplify √93

The prime factorization of 93 is 3 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √93 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 93, 3 and 31 appear an odd number of times, so √93 is irrational and 9.6436507610 is a rounded value.

Where √93 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √93 lies between 9 and 10. 93 is 12 above 81 and 7 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.6316 (0.13% low)
• Tangent from 9, i.e. 9 + 12 ÷ 18: 9.6667 (0.24% high)
• Tangent from 10, i.e. 10 − 7 ÷ 20: 9.6500 (0.07% high)

For √93 the tangent at 10 wins, missing by only 0.0063. Tangent estimates shine when the number sits close to a perfect square — here 93 is just 7 below 100.

Finding √93 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 93: following the tangent line down to zero simplifies to averaging x with 93 ÷ x.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √93 = 9.6436507610 to every decimal shown.

√93 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √93 the pattern is [9; 1, 1, 1, 4, 6, 4, 1, 1, 1, 18] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √93 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 − 93y² = 1. Its smallest solution in positive whole numbers is x = 12,151, y = 1,260.

√93 in geometry and everyday measurements

• A square room or garden bed covering 93 square feet measures about 9.64 ft (9 ft 8 in) along each wall.
• 93 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √93 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 8 box, because 2² + 5² + 8² = 93.

Square roots near √93 and related roots

• The cube root of 93 is about 4.530655.
• Four times the radicand doubles the root: √372 = 2 × √93 ≈ 19.287302.

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