Square Root of 297

√297 at a glance

Exact value

3√33

Decimal (10 places)

17.2336879396

Rounded

17.2 · 17.23 · 17.234

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.233688

Prime factorization

3³ × 11

Cube root

6.671940

How to simplify √297

Look for the largest perfect square that divides 297. Here it is 9 (3²), because 297 = 9 × 33 and 33 has no square factor left:

The prime factorization tells the same story: 297 = 3³ × 11. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 11 stays inside.

Check: (3√33)² = 3² × 33 = 9 × 33 = 297. As a decimal, 3√33 = 3 × 5.7445626465 ≈ 17.2336879396.

Where √297 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √297 lies between 17 and 18. 297 is 8 above 289 and 27 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.2286 (0.03% low)
• Tangent from 17, i.e. 17 + 8 ÷ 34: 17.2353 (0.01% high)
• Tangent from 18, i.e. 18 − 27 ÷ 36: 17.2500 (0.09% high)

For √297 the tangent at 17 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 297 is just 8 above 289.

Finding √297 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 297: following the tangent line down to zero simplifies to averaging x with 297 ÷ 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 √297 = 17.2336879396 to every decimal shown.

√297 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √297 the pattern is [17; 4, 3, 1, 1, 2, 1, 1, 3, 4, 34] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √297 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 − 297y² = 1. Its smallest solution in positive whole numbers is x = 48,599, y = 2,820.

√297 in geometry and everyday measurements

• A square patio or deck of 297 square feet is about 17.23 ft (17 ft 3 in) on each side, so edging all the way around takes 4 × √297 ≈ 68.9 ft.
• 297 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √297 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 14 box, because 1² + 10² + 14² = 297.
• Since √297 = 3√33, a length of √297 is exactly 3 copies of the length √33 laid end to end.

Square roots near √297 and related roots

• The cube root of 297 is about 6.671940.
• Squaring undoes the root: (√297)² = 297, while 297² = 88,209 — the number whose square root is 297.

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