Square Root of 532

√532 at a glance

Exact value

2√133

Decimal (10 places)

23.0651251893

Rounded

23.1 · 23.07 · 23.065

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.065125

Prime factorization

2² × 7 × 19

Cube root

8.102839

How to simplify √532

Look for the largest perfect square that divides 532. Here it is 4 (2²), because 532 = 4 × 133 and 133 has no square factor left:

The prime factorization tells the same story: 532 = 2² × 7 × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 19 stays inside.

Check: (2√133)² = 2² × 133 = 4 × 133 = 532. As a decimal, 2√133 = 2 × 11.5325625947 ≈ 23.0651251893.

Where √532 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √532 lies between 23 and 24. 532 is 3 above 529 and 44 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.0638 (0.01% low)
• Tangent from 23, i.e. 23 + 3 ÷ 46: 23.0652 (0% high)
• Tangent from 24, i.e. 24 − 44 ÷ 48: 23.0833 (0.08% high)

For √532 the tangent at 23 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 532 is just 3 above 529.

Finding √532 with the Babylonian method

Picture a rectangle with an area of 532 and one side x; the other side must be 532 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √532.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √532 = 23.0651251893 to every decimal shown.

√532 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √532 the pattern is [23; 15, 2, 1, 4, 2, 4, 1, 2, 15, 46] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √532 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 − 532y² = 1. Its smallest solution in positive whole numbers is x = 2,588,599, y = 112,230.

√532 in geometry and everyday measurements

• A square garage floor of 532 square feet measures about 23.07 ft (23 ft 1 in) per side, and its corner-to-corner diagonal is √1064 ≈ 32.6 ft.
• 532 is not a sum of two whole-number squares — the prime factor 7 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √532 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 12 × 18 box, because 8² + 12² + 18² = 532.
• Since √532 = 2√133, a length of √532 is exactly 2 copies of the length √133 laid end to end.

Square roots near √532 and related roots

• The cube root of 532 is about 8.102839.
• Because 532 = 4 × 133, the root is twice √133: 2 × 11.532563 ≈ 23.065125.

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