Square Root of 705

√705 at a glance

Exact value

√705

Decimal (10 places)

26.5518360947

Rounded

26.6 · 26.55 · 26.552

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.551836

Prime factorization

3 × 5 × 47

Cube root

8.900130

How to simplify √705

The prime factorization of 705 is 3 × 5 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √705 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 705, 3, 5 and 47 appear an odd number of times, so √705 is irrational and 26.5518360947 is a rounded value.

Where √705 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √705 lies between 26 and 27. 705 is 29 above 676 and 24 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.5472 (0.02% low)
• Tangent from 26, i.e. 26 + 29 ÷ 52: 26.5577 (0.02% high)
• Tangent from 27, i.e. 27 − 24 ÷ 54: 26.5556 (0.01% high)

For √705 the tangent at 27 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 705 is just 24 below 729.

Finding √705 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 705: following the tangent line down to zero simplifies to averaging x with 705 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √705 = 26.5518360947 to every decimal shown.

√705 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √705 the pattern is [26; 1, 1, 4, 3, 10, 3, 4, 1, 1, 52] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √705 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 − 705y² = 1. Its smallest solution in positive whole numbers is x = 237,161, y = 8,932.

√705 in geometry and everyday measurements

• 705 square feet is 65.5 m². Laid out as a square — a small house footprint or a lot — it is about 26.55 ft (26 ft 7 in) on a side.
• 705 is not a sum of two whole-number squares — the prime factor 3 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √705 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 26 box, because 2² + 5² + 26² = 705.

Square roots near √705 and related roots

• The cube root of 705 is about 8.900130.
• Squaring undoes the root: (√705)² = 705, while 705² = 497,025 — the number whose square root is 705.

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