Square Root of 73

√73 at a glance

Exact value

√73

Decimal (10 places)

8.5440037453

Rounded

8.5 · 8.54 · 8.544

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.544004

Prime factorization

73

Cube root

4.179339

How to simplify √73

73 is a prime number, so its only factors are 1 and 73. There is no perfect-square factor to pull out, which means √73 is already in its simplest radical form.

The square root of any prime is irrational. If √73 were a fraction a/b in lowest terms, then a² = 73b², so 73 would divide a — and then 73 would divide b too, contradicting “lowest terms.” That is why the decimal 8.5440037453 is only a rounded value.

Where √73 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √73 lies between 8 and 9. 73 is 9 above 64 and 8 below 81, so the root is closer to 9.

• Straight line between 64 and 81: 8.5294 (0.17% low)
• Tangent from 8, i.e. 8 + 9 ÷ 16: 8.5625 (0.22% high)
• Tangent from 9, i.e. 9 − 8 ÷ 18: 8.5556 (0.14% high)

For √73 the tangent at 9 wins, missing by only 0.0116. Tangent estimates shine when the number sits close to a perfect square — here 73 is just 8 below 81.

Finding √73 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 73: following the tangent line down to zero simplifies to averaging x with 73 ÷ 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 √73 = 8.5440037453 to every decimal shown.

√73 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √73 the pattern is [8; 1, 1, 5, 5, 1, 1, 16] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √73 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 − 73y² = 1. Its smallest solution in positive whole numbers is x = 2,281,249, y = 267,000. Because the period is odd, the equation with −1 on the right also has a solution: 1,068² − 73 × 125² = −1.

√73 in geometry and everyday measurements

• A square room or garden bed covering 73 square feet measures about 8.54 ft (8 ft 7 in) along each wall.
• 73 = 3² + 8², so by the Pythagorean theorem √73 is the diagonal of a 3 × 8 rectangle — and the distance between the points (0, 0) and (3, 8) on a grid.

Square roots near √73 and related roots

• The cube root of 73 is about 4.179339.
• Four times the radicand doubles the root: √292 = 2 × √73 ≈ 17.088007.

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