Square Root of 91

√91 at a glance

Exact value

√91

Decimal (10 places)

9.5393920142

Rounded

9.5 · 9.54 · 9.539

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.539392

Prime factorization

7 × 13

Cube root

4.497941

How to simplify √91

The prime factorization of 91 is 7 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √91 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 91, 7 and 13 appear an odd number of times, so √91 is irrational and 9.5393920142 is a rounded value.

Where √91 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √91 lies between 9 and 10. 91 is 10 above 81 and 9 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.5263 (0.14% low)
• Tangent from 9, i.e. 9 + 10 ÷ 18: 9.5556 (0.17% high)
• Tangent from 10, i.e. 10 − 9 ÷ 20: 9.5500 (0.11% high)

For √91 the tangent at 10 wins, missing by only 0.0106. Tangent estimates shine when the number sits close to a perfect square — here 91 is just 9 below 100.

Finding √91 with the Babylonian method

If a guess is too big, 91 ÷ 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√91) in one step.

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

The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √91 = 9.5393920142 to every decimal shown.

√91 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √91 the pattern is [9; 1, 1, 5, 1, 5, 1, 1, 18] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √91 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 − 91y² = 1. Its smallest solution in positive whole numbers is x = 1,574, y = 165.

√91 in geometry and everyday measurements

• A square room or garden bed covering 91 square feet measures about 9.54 ft (9 ft 6 in) along each wall.
• 91 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 √91 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 9 box, because 1² + 3² + 9² = 91.

Square roots near √91 and related roots

• The cube root of 91 is about 4.497941.
• Four times the radicand doubles the root: √364 = 2 × √91 ≈ 19.078784.

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