Square Root of 530

√530 at a glance

Exact value

√530

Decimal (10 places)

23.0217288664

Rounded

23.0 · 23.02 · 23.022

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.021729

Prime factorization

2 × 5 × 53

Cube root

8.092672

How to simplify √530

The prime factorization of 530 is 2 × 5 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √530 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 530, 2, 5 and 53 appear an odd number of times, so √530 is irrational and 23.0217288664 is a rounded value.

Where √530 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √530 lies between 23 and 24. 530 is 1 above 529 and 46 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.0213 (0% low)
• Tangent from 23, i.e. 23 + 1 ÷ 46: 23.0217 (0% high)
• Tangent from 24, i.e. 24 − 46 ÷ 48: 23.0417 (0.09% high)

For √530 the tangent at 23 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 530 is just 1 above 529.

Finding √530 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

Because the starting guess was already close, two steps are enough to match √530 = 23.0217288664 to every decimal shown.

√530 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √530 the pattern is [23; 46] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 530 is one more than a perfect square (23² + 1). A pattern that never ends is one more proof that √530 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 − 530y² = 1. Its smallest solution in positive whole numbers is x = 1,059, y = 46. Because the period is odd, the equation with −1 on the right also has a solution: 23² − 530 × 1² = −1.

√530 in geometry and everyday measurements

• A square garage floor of 530 square feet measures about 23.02 ft (23 ft) per side, and its corner-to-corner diagonal is √1060 ≈ 32.6 ft.
• 530 = 1² + 23² = 13² + 19², so by the Pythagorean theorem √530 is the diagonal of rectangles measuring 1 × 23 and 13 × 19 — and the distance between the points (0, 0) and (1, 23) on a grid.

Square roots near √530 and related roots

• The cube root of 530 is about 8.092672.
• Squaring undoes the root: (√530)² = 530, while 530² = 280,900 — the number whose square root is 530.

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