Square Root of 241

√241 at a glance

Exact value

√241

Decimal (10 places)

15.5241746963

Rounded

15.5 · 15.52 · 15.524

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.524175

Prime factorization

241

Cube root

6.223084

How to simplify √241

241 is a prime number, so its only factors are 1 and 241. There is no perfect-square factor to pull out, which means √241 is already in its simplest radical form.

The square root of any prime is irrational. If √241 were a fraction a/b in lowest terms, then a² = 241b², so 241 would divide a — and then 241 would divide b too, contradicting “lowest terms.” That is why the decimal 15.5241746963 is only a rounded value.

Where √241 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √241 lies between 15 and 16. 241 is 16 above 225 and 15 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.5161 (0.05% low)
• Tangent from 15, i.e. 15 + 16 ÷ 30: 15.5333 (0.06% high)
• Tangent from 16, i.e. 16 − 15 ÷ 32: 15.5313 (0.05% high)

For √241 the tangent at 16 wins, missing by only 0.0071. Tangent estimates shine when the number sits close to a perfect square — here 241 is just 15 below 256.

Finding √241 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 241: following the tangent line down to zero simplifies to averaging x with 241 ÷ x.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √241 = 15.5241746963 to every decimal shown.

√241 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √241 the pattern is [15; 1, 1, 9, 1, 5, 3, 3, 1, 1, 3, 3, 5, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √241 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 − 241y² = 1. Its smallest solution in positive whole numbers is x = 10,085,143,557,001,249, y = 649,641,205,044,600 — 17 digits for x, even though 241 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 71,011,068² − 241 × 4,574,225² = −1.

√241 in geometry and everyday measurements

• A square patio or deck of 241 square feet is about 15.52 ft (15 ft 6 in) on each side, so edging all the way around takes 4 × √241 ≈ 62.1 ft.
• 241 = 4² + 15², so by the Pythagorean theorem √241 is the diagonal of a 4 × 15 rectangle — and the distance between the points (0, 0) and (4, 15) on a grid.

Square roots near √241 and related roots

• The cube root of 241 is about 6.223084.
• Four times the radicand doubles the root: √964 = 2 × √241 ≈ 31.048349.

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