Square Root of 546

√546 at a glance

Exact value

√546

Decimal (10 places)

23.3666428911

Rounded

23.4 · 23.37 · 23.367

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.366643

Prime factorization

2 × 3 × 7 × 13

Cube root

8.173302

How to simplify √546

The prime factorization of 546 is 2 × 3 × 7 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √546 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 546, 2, 3, 7 and 13 appear an odd number of times, so √546 is irrational and 23.3666428911 is a rounded value.

Where √546 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √546 lies between 23 and 24. 546 is 17 above 529 and 30 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.3617 (0.02% low)
• Tangent from 23, i.e. 23 + 17 ÷ 46: 23.3696 (0.01% high)
• Tangent from 24, i.e. 24 − 30 ÷ 48: 23.3750 (0.04% high)

For √546 the tangent at 23 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 546 is just 17 above 529.

Finding √546 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):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √546 = 23.3666428911 to every decimal shown.

√546 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √546 the pattern is [23; 2, 1, 2, 1, 2, 46] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √546 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 − 546y² = 1. Its smallest solution in positive whole numbers is x = 701, y = 30.

√546 in geometry and everyday measurements

• A square garage floor of 546 square feet measures about 23.37 ft (23 ft 4 in) per side, and its corner-to-corner diagonal is √1092 ≈ 33 ft.
• 546 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √546 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 23 box, because 1² + 4² + 23² = 546.

Square roots near √546 and related roots

• The cube root of 546 is about 8.173302.
• Squaring undoes the root: (√546)² = 546, while 546² = 298,116 — the number whose square root is 546.

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