Square Root of 353

√353 at a glance

Exact value

√353

Decimal (10 places)

18.7882942281

Rounded

18.8 · 18.79 · 18.788

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.788294

Prime factorization

353

Cube root

7.067377

How to simplify √353

353 is a prime number, so its only factors are 1 and 353. There is no perfect-square factor to pull out, which means √353 is already in its simplest radical form.

The square root of any prime is irrational. If √353 were a fraction a/b in lowest terms, then a² = 353b², so 353 would divide a — and then 353 would divide b too, contradicting “lowest terms.” That is why the decimal 18.7882942281 is only a rounded value.

Where √353 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √353 lies between 18 and 19. 353 is 29 above 324 and 8 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.7838 (0.02% low)
• Tangent from 18, i.e. 18 + 29 ÷ 36: 18.8056 (0.09% high)
• Tangent from 19, i.e. 19 − 8 ÷ 38: 18.7895 (0.01% high)

For √353 the tangent at 19 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 353 is just 8 below 361.

Finding √353 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 353: following the tangent line down to zero simplifies to averaging x with 353 ÷ x.

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 √353 = 18.7882942281 to every decimal shown.

√353 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √353 the pattern is [18; 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √353 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 − 353y² = 1. Its smallest solution in positive whole numbers is x = 10,157,115,393, y = 540,608,704. Because the period is odd, the equation with −1 on the right also has a solution: 71,264² − 353 × 3,793² = −1.

√353 in geometry and everyday measurements

• A square patio or deck of 353 square feet is about 18.79 ft (18 ft 9 in) on each side, so edging all the way around takes 4 × √353 ≈ 75.2 ft.
• 353 = 8² + 17², so by the Pythagorean theorem √353 is the diagonal of a 8 × 17 rectangle — and the distance between the points (0, 0) and (8, 17) on a grid.

Square roots near √353 and related roots

• The cube root of 353 is about 7.067377.
• Squaring undoes the root: (√353)² = 353, while 353² = 124,609 — the number whose square root is 353.

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