Square Root of 281

√281 at a glance

Exact value

√281

Decimal (10 places)

16.7630546142

Rounded

16.8 · 16.76 · 16.763

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.763055

Prime factorization

281

Cube root

6.549912

How to simplify √281

281 is a prime number, so its only factors are 1 and 281. There is no perfect-square factor to pull out, which means √281 is already in its simplest radical form.

The square root of any prime is irrational. If √281 were a fraction a/b in lowest terms, then a² = 281b², so 281 would divide a — and then 281 would divide b too, contradicting “lowest terms.” That is why the decimal 16.7630546142 is only a rounded value.

Where √281 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √281 lies between 16 and 17. 281 is 25 above 256 and 8 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.7576 (0.03% low)
• Tangent from 16, i.e. 16 + 25 ÷ 32: 16.7813 (0.11% high)
• Tangent from 17, i.e. 17 − 8 ÷ 34: 16.7647 (0.01% high)

For √281 the tangent at 17 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 281 is just 8 below 289.

Finding √281 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 281: following the tangent line down to zero simplifies to averaging x with 281 ÷ x.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √281 = 16.7630546142 to every decimal shown.

√281 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √281 the pattern is [16; 1, 3, 4, 1, 1, 6, 6, 1, 1, 4, 3, 1, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √281 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 − 281y² = 1. Its smallest solution in positive whole numbers is x = 2,262,200,630,049, y = 134,951,575,480 — 13 digits for x, even though 281 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: 1,063,532² − 281 × 63,445² = −1.

√281 in geometry and everyday measurements

• A square patio or deck of 281 square feet is about 16.76 ft (16 ft 9 in) on each side, so edging all the way around takes 4 × √281 ≈ 67.1 ft.
• 281 = 5² + 16², so by the Pythagorean theorem √281 is the diagonal of a 5 × 16 rectangle — and the distance between the points (0, 0) and (5, 16) on a grid.

Square roots near √281 and related roots

• The cube root of 281 is about 6.549912.
• Squaring undoes the root: (√281)² = 281, while 281² = 78,961 — the number whose square root is 281.

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