Square Root of 352

√352 at a glance

Exact value

4√22

Decimal (10 places)

18.7616630393

Rounded

18.8 · 18.76 · 18.762

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.761663

Prime factorization

2⁵ × 11

Cube root

7.060697

How to simplify √352

Look for the largest perfect square that divides 352. Here it is 16 (4²), because 352 = 16 × 22 and 22 has no square factor left:

The prime factorization tells the same story: 352 = 2⁵ × 11. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 11 stays inside.

352 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √352 = 2√88, and √88 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√22)² = 4² × 22 = 16 × 22 = 352. As a decimal, 4√22 = 4 × 4.6904157598 ≈ 18.7616630393.

Where √352 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √352 lies between 18 and 19. 352 is 28 above 324 and 9 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.7568 (0.03% low)
• Tangent from 18, i.e. 18 + 28 ÷ 36: 18.7778 (0.09% high)
• Tangent from 19, i.e. 19 − 9 ÷ 38: 18.7632 (0.01% high)

For √352 the tangent at 19 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 352 is just 9 below 361.

Finding √352 with the Babylonian method

Picture a rectangle with an area of 352 and one side x; the other side must be 352 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √352.

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 √352 = 18.7616630393 to every decimal shown.

√352 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √352 the pattern is [18; 1, 3, 5, 9, 5, 3, 1, 36] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √352 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 − 352y² = 1. Its smallest solution in positive whole numbers is x = 77,617, y = 4,137.

√352 in geometry and everyday measurements

• A square patio or deck of 352 square feet is about 18.76 ft (18 ft 9 in) on each side, so edging all the way around takes 4 × √352 ≈ 75 ft.
• 352 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √352 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 12 × 12 box, because 8² + 12² + 12² = 352.
• Since √352 = 4√22, a length of √352 is exactly 4 copies of the length √22 laid end to end.

Square roots near √352 and related roots

• The cube root of 352 is about 7.060697.
• Because 352 = 4 × 88, the root is twice √88: 2 × 9.380832 ≈ 18.761663.

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