Square Root of 735

√735 at a glance

Exact value

7√15

Decimal (10 places)

27.1108834235

Rounded

27.1 · 27.11 · 27.111

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.110883

Prime factorization

3 × 5 × 7²

Cube root

9.024624

How to simplify √735

Look for the largest perfect square that divides 735. Here it is 49 (7²), because 735 = 49 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 735 = 3 × 5 × 7². Each pair of equal primes leaves the radical as one factor, so 7 comes out and 3 × 5 stays inside.

Check: (7√15)² = 7² × 15 = 49 × 15 = 735. As a decimal, 7√15 = 7 × 3.8729833462 ≈ 27.1108834235.

Where √735 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √735 lies between 27 and 28. 735 is 6 above 729 and 49 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.1091 (0.01% low)
• Tangent from 27, i.e. 27 + 6 ÷ 54: 27.1111 (0% high)
• Tangent from 28, i.e. 28 − 49 ÷ 56: 27.1250 (0.05% high)

For √735 the tangent at 27 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 735 is just 6 above 729.

Finding √735 with the Babylonian method

If a guess is too big, 735 ÷ 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√735) in one step.

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 √735 = 27.1108834235 to every decimal shown.

√735 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √735 the pattern is [27; 9, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √735 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 − 735y² = 1. Its smallest solution in positive whole numbers is x = 244, y = 9.

√735 in geometry and everyday measurements

• 735 square feet is 68.3 m². Laid out as a square — a small house footprint or a lot — it is about 27.11 ft (27 ft 1 in) on a side.
• 735 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √735 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √735 as its space diagonal.
• Since √735 = 7√15, a length of √735 is exactly 7 copies of the length √15 laid end to end.

Square roots near √735 and related roots

• The cube root of 735 is about 9.024624.
• Squaring undoes the root: (√735)² = 735, while 735² = 540,225 — the number whose square root is 735.

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