Square Root of 701

√701 at a glance

Exact value

√701

Decimal (10 places)

26.4764045897

Rounded

26.5 · 26.48 · 26.476

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.476405

Prime factorization

701

Cube root

8.883266

How to simplify √701

701 is a prime number, so its only factors are 1 and 701. There is no perfect-square factor to pull out, which means √701 is already in its simplest radical form.

The square root of any prime is irrational. If √701 were a fraction a/b in lowest terms, then a² = 701b², so 701 would divide a — and then 701 would divide b too, contradicting “lowest terms.” That is why the decimal 26.4764045897 is only a rounded value.

Where √701 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √701 lies between 26 and 27. 701 is 25 above 676 and 28 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.4717 (0.02% low)
• Tangent from 26, i.e. 26 + 25 ÷ 52: 26.4808 (0.02% high)
• Tangent from 27, i.e. 27 − 28 ÷ 54: 26.4815 (0.02% high)

For √701 the tangent at 26 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 701 is just 25 above 676.

Finding √701 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 701: following the tangent line down to zero simplifies to averaging x with 701 ÷ x.

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 √701 = 26.4764045897 to every decimal shown.

√701 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √701 the pattern is [26; 2, 10, 10, 2, 52] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √701 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 − 701y² = 1. Its smallest solution in positive whole numbers is x = 277,631,049, y = 10,485,980. Because the period is odd, the equation with −1 on the right also has a solution: 11,782² − 701 × 445² = −1.

√701 in geometry and everyday measurements

• 701 square feet is 65.1 m². Laid out as a square — a small house footprint or a lot — it is about 26.48 ft (26 ft 6 in) on a side.
• 701 = 5² + 26², so by the Pythagorean theorem √701 is the diagonal of a 5 × 26 rectangle — and the distance between the points (0, 0) and (5, 26) on a grid.

Square roots near √701 and related roots

• The cube root of 701 is about 8.883266.
• Squaring undoes the root: (√701)² = 701, while 701² = 491,401 — the number whose square root is 701.

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