Square Root of 232

√232 at a glance

Exact value

2√58

Decimal (10 places)

15.2315462117

Rounded

15.2 · 15.23 · 15.232

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.231546

Prime factorization

2³ × 29

Cube root

6.144634

How to simplify √232

Look for the largest perfect square that divides 232. Here it is 4 (2²), because 232 = 4 × 58 and 58 has no square factor left:

The prime factorization tells the same story: 232 = 2³ × 29. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 29 stays inside.

Check: (2√58)² = 2² × 58 = 4 × 58 = 232. As a decimal, 2√58 = 2 × 7.6157731059 ≈ 15.2315462117.

Where √232 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √232 lies between 15 and 16. 232 is 7 above 225 and 24 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.2258 (0.04% low)
• Tangent from 15, i.e. 15 + 7 ÷ 30: 15.2333 (0.01% high)
• Tangent from 16, i.e. 16 − 24 ÷ 32: 15.2500 (0.12% high)

For √232 the tangent at 15 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 232 is just 7 above 225.

Finding √232 with the Babylonian method

Picture a rectangle with an area of 232 and one side x; the other side must be 232 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √232.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √232 = 15.2315462117 to every decimal shown.

√232 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √232 the pattern is [15; 4, 3, 7, 3, 4, 30] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √232 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 − 232y² = 1. Its smallest solution in positive whole numbers is x = 19,603, y = 1,287.

√232 in geometry and everyday measurements

• A square patio or deck of 232 square feet is about 15.23 ft (15 ft 3 in) on each side, so edging all the way around takes 4 × √232 ≈ 60.9 ft.
• 232 = 6² + 14², so by the Pythagorean theorem √232 is the diagonal of a 6 × 14 rectangle — and the distance between the points (0, 0) and (6, 14) on a grid.
• Since √232 = 2√58, a length of √232 is exactly 2 copies of the length √58 laid end to end.

Square roots near √232 and related roots

• The cube root of 232 is about 6.144634.
• Four times the radicand doubles the root: √928 = 2 × √232 ≈ 30.463092.

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