Square Root of 716

√716 at a glance

Exact value

2√179

Decimal (10 places)

26.7581763205

Rounded

26.8 · 26.76 · 26.758

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.758176

Prime factorization

2² × 179

Cube root

8.946181

How to simplify √716

Look for the largest perfect square that divides 716. Here it is 4 (2²), because 716 = 4 × 179 and 179 has no square factor left:

The prime factorization tells the same story: 716 = 2² × 179. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 179 stays inside.

Check: (2√179)² = 2² × 179 = 4 × 179 = 716. As a decimal, 2√179 = 2 × 13.3790881603 ≈ 26.7581763205.

Where √716 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √716 lies between 26 and 27. 716 is 40 above 676 and 13 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.7547 (0.01% low)
• Tangent from 26, i.e. 26 + 40 ÷ 52: 26.7692 (0.04% high)
• Tangent from 27, i.e. 27 − 13 ÷ 54: 26.7593 (0% high)

For √716 the tangent at 27 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 716 is just 13 below 729.

Finding √716 with the Babylonian method

Picture a rectangle with an area of 716 and one side x; the other side must be 716 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √716.

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 √716 = 26.7581763205 to every decimal shown.

√716 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √716 the pattern is [26; 1, 3, 7, 2, 1, 1, 6, 10, 1, 1, 4, 2, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √716 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 − 716y² = 1. Its smallest solution in positive whole numbers is x = 35,115,719,688,199, y = 1,312,336,060,110 — 14 digits for x, even though 716 is small, which is what makes Pell’s equation famous.

√716 in geometry and everyday measurements

• 716 square feet is 66.5 m². Laid out as a square — a small house footprint or a lot — it is about 26.76 ft (26 ft 9 in) on a side.
• 716 is not a sum of two whole-number squares — the prime factor 179 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √716 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 26 box, because 2² + 6² + 26² = 716.
• Since √716 = 2√179, a length of √716 is exactly 2 copies of the length √179 laid end to end.

Square roots near √716 and related roots

• The cube root of 716 is about 8.946181.
• Because 716 = 4 × 179, the root is twice √179: 2 × 13.379088 ≈ 26.758176.

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