Square Root of 89

√89 at a glance

Exact value

√89

Decimal (10 places)

9.4339811321

Rounded

9.4 · 9.43 · 9.434

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.433981

Prime factorization

89

Cube root

4.464745

How to simplify √89

89 is a prime number, so its only factors are 1 and 89. There is no perfect-square factor to pull out, which means √89 is already in its simplest radical form.

The square root of any prime is irrational. If √89 were a fraction a/b in lowest terms, then a² = 89b², so 89 would divide a — and then 89 would divide b too, contradicting “lowest terms.” That is why the decimal 9.4339811321 is only a rounded value.

Where √89 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √89 lies between 9 and 10. 89 is 8 above 81 and 11 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.4211 (0.14% low)
• Tangent from 9, i.e. 9 + 8 ÷ 18: 9.4444 (0.11% high)
• Tangent from 10, i.e. 10 − 11 ÷ 20: 9.4500 (0.17% high)

For √89 the tangent at 9 wins, missing by only 0.0105. Tangent estimates shine when the number sits close to a perfect square — here 89 is just 8 above 81.

Finding √89 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 89: following the tangent line down to zero simplifies to averaging x with 89 ÷ x.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √89 = 9.4339811321 to every decimal shown.

√89 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √89 the pattern is [9; 2, 3, 3, 2, 18] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √89 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 − 89y² = 1. Its smallest solution in positive whole numbers is x = 500,001, y = 53,000. Because the period is odd, the equation with −1 on the right also has a solution: 500² − 89 × 53² = −1.

√89 in geometry and everyday measurements

• A square room or garden bed covering 89 square feet measures about 9.43 ft (9 ft 5 in) along each wall.
• 89 = 5² + 8², so by the Pythagorean theorem √89 is the diagonal of a 5 × 8 rectangle — and the distance between the points (0, 0) and (5, 8) on a grid.

Square roots near √89 and related roots

• The cube root of 89 is about 4.464745.
• Four times the radicand doubles the root: √356 = 2 × √89 ≈ 18.867962.

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