Square Root of 713

√713 at a glance

Exact value

√713

Decimal (10 places)

26.7020598456

Rounded

26.7 · 26.70 · 26.702

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.702060

Prime factorization

23 × 31

Cube root

8.933669

How to simplify √713

The prime factorization of 713 is 23 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √713 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 713, 23 and 31 appear an odd number of times, so √713 is irrational and 26.7020598456 is a rounded value.

Where √713 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √713 lies between 26 and 27. 713 is 37 above 676 and 16 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.6981 (0.01% low)
• Tangent from 26, i.e. 26 + 37 ÷ 52: 26.7115 (0.04% high)
• Tangent from 27, i.e. 27 − 16 ÷ 54: 26.7037 (0.01% high)

For √713 the tangent at 27 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 713 is just 16 below 729.

Finding √713 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 713: following the tangent line down to zero simplifies to averaging x with 713 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √713 = 26.7020598456 to every decimal shown.

√713 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √713 the pattern is [26; 1, 2, 2, 1, 4, 6, 2, 6, 4, 1, 2, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √713 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 − 713y² = 1. Its smallest solution in positive whole numbers is x = 5,286,367, y = 197,976.

√713 in geometry and everyday measurements

• 713 square feet is 66.2 m². Laid out as a square — a small house footprint or a lot — it is about 26.7 ft (26 ft 8 in) on a side.
• 713 is not a sum of two whole-number squares — the prime factor 23 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √713 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 26 box, because 1² + 6² + 26² = 713.

Square roots near √713 and related roots

• The cube root of 713 is about 8.933669.
• Squaring undoes the root: (√713)² = 713, while 713² = 508,369 — the number whose square root is 713.

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