Square Root of 303

√303 at a glance

Exact value

√303

Decimal (10 places)

17.4068951855

Rounded

17.4 · 17.41 · 17.407

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.406895

Prime factorization

3 × 101

Cube root

6.716570

How to simplify √303

The prime factorization of 303 is 3 × 101. Every prime appears only once, so there is no pair to bring outside the radical — √303 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 303, 3 and 101 appear an odd number of times, so √303 is irrational and 17.4068951855 is a rounded value.

Where √303 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √303 lies between 17 and 18. 303 is 14 above 289 and 21 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.4000 (0.04% low)
• Tangent from 17, i.e. 17 + 14 ÷ 34: 17.4118 (0.03% high)
• Tangent from 18, i.e. 18 − 21 ÷ 36: 17.4167 (0.06% high)

For √303 the tangent at 17 wins, missing by only 0.0049. Tangent estimates shine when the number sits close to a perfect square — here 303 is just 14 above 289.

Finding √303 with the Babylonian method

If a guess is too big, 303 ÷ 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√303) in one step.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √303 = 17.4068951855 to every decimal shown.

√303 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √303 the pattern is [17; 2, 2, 5, 2, 2, 34] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √303 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 − 303y² = 1. Its smallest solution in positive whole numbers is x = 2,524, y = 145.

√303 in geometry and everyday measurements

• A square patio or deck of 303 square feet is about 17.41 ft (17 ft 5 in) on each side, so edging all the way around takes 4 × √303 ≈ 69.6 ft.
• 303 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √303 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 √303 as its space diagonal.

Square roots near √303 and related roots

• The cube root of 303 is about 6.716570.
• Squaring undoes the root: (√303)² = 303, while 303² = 91,809 — the number whose square root is 303.

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