Square Root of 702

√702 at a glance

Exact value

3√78

Decimal (10 places)

26.4952825990

Rounded

26.5 · 26.50 · 26.495

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.495283

Prime factorization

2 × 3³ × 13

Cube root

8.887488

How to simplify √702

Look for the largest perfect square that divides 702. Here it is 9 (3²), because 702 = 9 × 78 and 78 has no square factor left:

The prime factorization tells the same story: 702 = 2 × 3³ × 13. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 3 × 13 stays inside.

Check: (3√78)² = 3² × 78 = 9 × 78 = 702. As a decimal, 3√78 = 3 × 8.8317608663 ≈ 26.4952825990.

Where √702 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √702 lies between 26 and 27. 702 is 26 above 676 and 27 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.4906 (0.02% low)
• Tangent from 26, i.e. 26 + 26 ÷ 52: 26.5000 (0.02% high)
• Tangent from 27, i.e. 27 − 27 ÷ 54: 26.5000 (0.02% high)

For √702 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √702 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √702 = 26.4952825990 to every decimal shown.

√702 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √702 the pattern is [26; 2, 52] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √702 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 − 702y² = 1. Its smallest solution in positive whole numbers is x = 53, y = 2.

√702 in geometry and everyday measurements

• 702 square feet is 65.2 m². Laid out as a square — a small house footprint or a lot — it is about 26.5 ft (26 ft 6 in) on a side.
• 702 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 √702 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 26 box, because 1² + 5² + 26² = 702.
• Since √702 = 3√78, a length of √702 is exactly 3 copies of the length √78 laid end to end.

Square roots near √702 and related roots

• The cube root of 702 is about 8.887488.
• Squaring undoes the root: (√702)² = 702, while 702² = 492,804 — the number whose square root is 702.

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