Square Root of 533

√533 at a glance

Exact value

√533

Decimal (10 places)

23.0867927612

Rounded

23.1 · 23.09 · 23.087

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.086793

Prime factorization

13 × 41

Cube root

8.107913

How to simplify √533

The prime factorization of 533 is 13 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √533 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 533, 13 and 41 appear an odd number of times, so √533 is irrational and 23.0867927612 is a rounded value.

Where √533 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √533 lies between 23 and 24. 533 is 4 above 529 and 43 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.0851 (0.01% low)
• Tangent from 23, i.e. 23 + 4 ÷ 46: 23.0870 (0% high)
• Tangent from 24, i.e. 24 − 43 ÷ 48: 23.1042 (0.08% high)

For √533 the tangent at 23 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 533 is just 4 above 529.

Finding √533 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 533: following the tangent line down to zero simplifies to averaging x with 533 ÷ x.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √533 = 23.0867927612 to every decimal shown.

√533 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √533 the pattern is [23; 11, 1, 1, 11, 46] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √533 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 − 533y² = 1. Its smallest solution in positive whole numbers is x = 74,859,849, y = 3,242,540. Because the period is odd, the equation with −1 on the right also has a solution: 6,118² − 533 × 265² = −1.

√533 in geometry and everyday measurements

• A square garage floor of 533 square feet measures about 23.09 ft (23 ft 1 in) per side, and its corner-to-corner diagonal is √1066 ≈ 32.6 ft.
• 533 = 2² + 23² = 7² + 22², so by the Pythagorean theorem √533 is the diagonal of rectangles measuring 2 × 23 and 7 × 22 — and the distance between the points (0, 0) and (2, 23) on a grid.

Square roots near √533 and related roots

• The cube root of 533 is about 8.107913.
• Squaring undoes the root: (√533)² = 533, while 533² = 284,089 — the number whose square root is 533.

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