Square Root of 351

√351 at a glance

Exact value

3√39

Decimal (10 places)

18.7349939952

Rounded

18.7 · 18.73 · 18.735

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.734994

Prime factorization

3³ × 13

Cube root

7.054004

How to simplify √351

Look for the largest perfect square that divides 351. Here it is 9 (3²), because 351 = 9 × 39 and 39 has no square factor left:

The prime factorization tells the same story: 351 = 3³ × 13. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 13 stays inside.

Check: (3√39)² = 3² × 39 = 9 × 39 = 351. As a decimal, 3√39 = 3 × 6.2449979984 ≈ 18.7349939952.

Where √351 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √351 lies between 18 and 19. 351 is 27 above 324 and 10 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.7297 (0.03% low)
• Tangent from 18, i.e. 18 + 27 ÷ 36: 18.7500 (0.08% high)
• Tangent from 19, i.e. 19 − 10 ÷ 38: 18.7368 (0.01% high)

For √351 the tangent at 19 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 351 is just 10 below 361.

Finding √351 with the Babylonian method

If a guess is too big, 351 ÷ 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√351) in one step.

Start from the nearest whole number, 19 (19² = 361):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √351 = 18.7349939952 to every decimal shown.

√351 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √351 the pattern is [18; 1, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 36] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √351 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 − 351y² = 1. Its smallest solution in positive whole numbers is x = 62,425, y = 3,332.

√351 in geometry and everyday measurements

• A square patio or deck of 351 square feet is about 18.73 ft (18 ft 9 in) on each side, so edging all the way around takes 4 × √351 ≈ 74.9 ft.
• 351 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 √351 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 √351 as its space diagonal.
• Since √351 = 3√39, a length of √351 is exactly 3 copies of the length √39 laid end to end.

Square roots near √351 and related roots

• The cube root of 351 is about 7.054004.
• Squaring undoes the root: (√351)² = 351, while 351² = 123,201 — the number whose square root is 351.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.