Square Root of 733

√733 at a glance

Exact value

√733

Decimal (10 places)

27.0739727414

Rounded

27.1 · 27.07 · 27.074

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.073973

Prime factorization

733

Cube root

9.016431

How to simplify √733

733 is a prime number, so its only factors are 1 and 733. There is no perfect-square factor to pull out, which means √733 is already in its simplest radical form.

The square root of any prime is irrational. If √733 were a fraction a/b in lowest terms, then a² = 733b², so 733 would divide a — and then 733 would divide b too, contradicting “lowest terms.” That is why the decimal 27.0739727414 is only a rounded value.

Where √733 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √733 lies between 27 and 28. 733 is 4 above 729 and 51 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.0727 (0% low)
• Tangent from 27, i.e. 27 + 4 ÷ 54: 27.0741 (0% high)
• Tangent from 28, i.e. 28 − 51 ÷ 56: 27.0893 (0.06% high)

For √733 the tangent at 27 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 733 is just 4 above 729.

Finding √733 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 733: following the tangent line down to zero simplifies to averaging x with 733 ÷ x.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √733 = 27.0739727414 to every decimal shown.

√733 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √733 the pattern is [27; 13, 1, 1, 13, 54] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √733 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 − 733y² = 1. Its smallest solution in positive whole numbers is x = 195,307,849, y = 7,213,860. Because the period is odd, the equation with −1 on the right also has a solution: 9,882² − 733 × 365² = −1.

√733 in geometry and everyday measurements

• 733 square feet is 68.1 m². Laid out as a square — a small house footprint or a lot — it is about 27.07 ft (27 ft 1 in) on a side.
• 733 = 2² + 27², so by the Pythagorean theorem √733 is the diagonal of a 2 × 27 rectangle — and the distance between the points (0, 0) and (2, 27) on a grid.

Square roots near √733 and related roots

• The cube root of 733 is about 9.016431.
• Squaring undoes the root: (√733)² = 733, while 733² = 537,289 — the number whose square root is 733.

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