Square Root of 336

√336 at a glance

Exact value

4√21

Decimal (10 places)

18.3303027798

Rounded

18.3 · 18.33 · 18.330

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.330303

Prime factorization

2⁴ × 3 × 7

Cube root

6.952053

How to simplify √336

Look for the largest perfect square that divides 336. Here it is 16 (4²), because 336 = 16 × 21 and 21 has no square factor left:

The prime factorization tells the same story: 336 = 2⁴ × 3 × 7. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 3 × 7 stays inside.

336 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √336 = 2√84, and √84 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√21)² = 4² × 21 = 16 × 21 = 336. As a decimal, 4√21 = 4 × 4.582575695 ≈ 18.3303027798.

Where √336 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √336 lies between 18 and 19. 336 is 12 above 324 and 25 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.3243 (0.03% low)
• Tangent from 18, i.e. 18 + 12 ÷ 36: 18.3333 (0.02% high)
• Tangent from 19, i.e. 19 − 25 ÷ 38: 18.3421 (0.06% high)

For √336 the tangent at 18 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 336 is just 12 above 324.

Finding √336 with the Babylonian method

Picture a rectangle with an area of 336 and one side x; the other side must be 336 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √336.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √336 = 18.3303027798 to every decimal shown.

√336 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √336 the pattern is [18; 3, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √336 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 − 336y² = 1. Its smallest solution in positive whole numbers is x = 55, y = 3.

√336 in geometry and everyday measurements

• A square patio or deck of 336 square feet is about 18.33 ft (18 ft 4 in) on each side, so edging all the way around takes 4 × √336 ≈ 73.3 ft.
• 336 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √336 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 16 box, because 4² + 8² + 16² = 336.
• Since √336 = 4√21, a length of √336 is exactly 4 copies of the length √21 laid end to end.

Square roots near √336 and related roots

• The cube root of 336 is about 6.952053.
• Because 336 = 4 × 84, the root is twice √84: 2 × 9.165151 ≈ 18.330303.

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