Square Root of 61

√61 at a glance

Exact value

√61

Decimal (10 places)

7.8102496759

Rounded

7.8 · 7.81 · 7.810

Perfect square?

No — between 7² and 8²

Rational?

Irrational

Both square roots

±7.810250

Prime factorization

61

Cube root

3.936497

How to simplify √61

61 is a prime number, so its only factors are 1 and 61. There is no perfect-square factor to pull out, which means √61 is already in its simplest radical form.

The square root of any prime is irrational. If √61 were a fraction a/b in lowest terms, then a² = 61b², so 61 would divide a — and then 61 would divide b too, contradicting “lowest terms.” That is why the decimal 7.8102496759 is only a rounded value.

Where √61 sits between perfect squares

49 = 7² and 64 = 8² are the nearest perfect squares, so √61 lies between 7 and 8. 61 is 12 above 49 and 3 below 64, so the root is closer to 8.

• Straight line between 49 and 64: 7.8000 (0.13% low)
• Tangent from 7, i.e. 7 + 12 ÷ 14: 7.8571 (0.6% high)
• Tangent from 8, i.e. 8 − 3 ÷ 16: 7.8125 (0.03% high)

For √61 the tangent at 8 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 61 is just 3 below 64.

Finding √61 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 61: following the tangent line down to zero simplifies to averaging x with 61 ÷ x.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √61 = 7.8102496759 to every decimal shown.

√61 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √61 the pattern is [7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √61 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 − 61y² = 1. Its smallest solution in positive whole numbers is x = 1,766,319,049, y = 226,153,980. Because the period is odd, the equation with −1 on the right also has a solution: 29,718² − 61 × 3,805² = −1.

√61 in geometry and everyday measurements

• A square room or garden bed covering 61 square feet measures about 7.81 ft (7 ft 10 in) along each wall.
• 61 = 5² + 6², so by the Pythagorean theorem √61 is the diagonal of a 5 × 6 rectangle — and the distance between the points (0, 0) and (5, 6) on a grid.

Square roots near √61 and related roots

• The cube root of 61 is about 3.936497.
• Four times the radicand doubles the root: √244 = 2 × √61 ≈ 15.620499.

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