Square Root of 287

√287 at a glance

Exact value

√287

Decimal (10 places)

16.9410743461

Rounded

16.9 · 16.94 · 16.941

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.941074

Prime factorization

7 × 41

Cube root

6.596202

How to simplify √287

The prime factorization of 287 is 7 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √287 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 287, 7 and 41 appear an odd number of times, so √287 is irrational and 16.9410743461 is a rounded value.

Where √287 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √287 lies between 16 and 17. 287 is 31 above 256 and 2 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.9394 (0.01% low)
• Tangent from 16, i.e. 16 + 31 ÷ 32: 16.9688 (0.16% high)
• Tangent from 17, i.e. 17 − 2 ÷ 34: 16.9412 (0% high)

For √287 the tangent at 17 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 287 is just 2 below 289.

Finding √287 with the Babylonian method

If a guess is too big, 287 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√287) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √287 = 16.9410743461 to every decimal shown.

√287 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √287 the pattern is [16; 1, 15, 1, 32] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √287 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 − 287y² = 1. Its smallest solution in positive whole numbers is x = 288, y = 17.

√287 in geometry and everyday measurements

• A square patio or deck of 287 square feet is about 16.94 ft (16 ft 11 in) on each side, so edging all the way around takes 4 × √287 ≈ 67.8 ft.
• 287 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √287 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √287 as its space diagonal.

Square roots near √287 and related roots

• The cube root of 287 is about 6.596202.
• Squaring undoes the root: (√287)² = 287, while 287² = 82,369 — the number whose square root is 287.

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