Square Root of 712

√712 at a glance

Exact value

2√178

Decimal (10 places)

26.6833281283

Rounded

26.7 · 26.68 · 26.683

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.683328

Prime factorization

2³ × 89

Cube root

8.929490

How to simplify √712

Look for the largest perfect square that divides 712. Here it is 4 (2²), because 712 = 4 × 178 and 178 has no square factor left:

The prime factorization tells the same story: 712 = 2³ × 89. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 89 stays inside.

Check: (2√178)² = 2² × 178 = 4 × 178 = 712. As a decimal, 2√178 = 2 × 13.3416640641 ≈ 26.6833281283.

Where √712 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √712 lies between 26 and 27. 712 is 36 above 676 and 17 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.6792 (0.02% low)
• Tangent from 26, i.e. 26 + 36 ÷ 52: 26.6923 (0.03% high)
• Tangent from 27, i.e. 27 − 17 ÷ 54: 26.6852 (0.01% high)

For √712 the tangent at 27 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 712 is just 17 below 729.

Finding √712 with the Babylonian method

Picture a rectangle with an area of 712 and one side x; the other side must be 712 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √712.

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 √712 = 26.6833281283 to every decimal shown.

√712 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √712 the pattern is [26; 1, 2, 6, 2, 1, 52] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √712 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 − 712y² = 1. Its smallest solution in positive whole numbers is x = 1,601, y = 60.

√712 in geometry and everyday measurements

• 712 square feet is 66.1 m². Laid out as a square — a small house footprint or a lot — it is about 26.68 ft (26 ft 8 in) on a side.
• 712 = 6² + 26², so by the Pythagorean theorem √712 is the diagonal of a 6 × 26 rectangle — and the distance between the points (0, 0) and (6, 26) on a grid.
• Since √712 = 2√178, a length of √712 is exactly 2 copies of the length √178 laid end to end.

Square roots near √712 and related roots

• The cube root of 712 is about 8.929490.
• Because 712 = 4 × 178, the root is twice √178: 2 × 13.341664 ≈ 26.683328.

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