√543 at a glance
- Exact value
- √543
- Decimal (10 places)
- 23.3023603955
- Rounded
- 23.3 · 23.30 · 23.302
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.302360
- Prime factorization
- 3 × 181
- Cube root
- 8.158305
How to simplify √543
The prime factorization of 543 is 3 × 181. Every prime appears only once, so there is no pair to bring outside the radical — √543 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 543, 3 and 181 appear an odd number of times, so √543 is irrational and 23.3023603955 is a rounded value.
Where √543 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √543 lies between 23 and 24. 543 is 14 above 529 and 33 below 576, so the root is closer to 23.
- Straight line between 529 and 576: 23.2979 (0.02% low)
- Tangent from 23, i.e. 23 + 14 ÷ 46: 23.3043 (0.01% high)
- Tangent from 24, i.e. 24 − 33 ÷ 48: 23.3125 (0.04% high)
For √543 the tangent at 23 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 543 is just 14 above 529.
Finding √543 with the Babylonian method
If a guess is too big, 543 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√543) in one step.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 543 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 23.6086956522 | 23.3043478261 | 2 |
| 2 | 23.3043478261 | 23.3003731343 | 23.3023604802 | 7 |
| 3 | 23.3023604802 | 23.3023603107 | 23.3023603955 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √543 = 23.3023603955 to every decimal shown.
√543 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √543 the pattern is [23; 3, 3, 3, 1, 14, 1, 3, 3, 3, 46] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √543 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 23/1 | 23.0000000000 | 3.0 × 10⁻¹ |
| 70/3 | 23.3333333333 | 3.1 × 10⁻² |
| 233/10 | 23.3000000000 | 2.4 × 10⁻³ |
| 769/33 | 23.3030303030 | 6.7 × 10⁻⁴ |
| 1,002/43 | 23.3023255814 | 3.5 × 10⁻⁵ |
| 14,797/635 | 23.3023622047 | 1.8 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 543y² = 1. Its smallest solution in positive whole numbers is x = 669,337, y = 28,724.
√543 in geometry and everyday measurements
- A square garage floor of 543 square feet measures about 23.3 ft (23 ft 4 in) per side, and its corner-to-corner diagonal is √1086 ≈ 33 ft.
- 543 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √543 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √543 as its space diagonal.
Square roots near √543 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √540 | 6√15 | 23.2379 | No |
| √541 | √541 | 23.2594 | No |
| √542 | √542 | 23.2809 | No |
| √543 | √543 | 23.3024 | No |
| √544 | 4√34 | 23.3238 | No |
| √545 | √545 | 23.3452 | No |
| √546 | √546 | 23.3666 | No |
- The cube root of 543 is about 8.158305.
- Squaring undoes the root: (√543)² = 543, while 543² = 294,849 — the number whose square root is 543.
Frequently asked questions
What is the square root of 543?
The square root of 543 is √543, about 23.3023603955. The negative root, −23.302360, also squares to 543.
Is the square root of 543 rational or irrational?
Irrational. 543 is not a perfect square — it falls between 529 and 576 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √543 be simplified?
No. 543 = 3 × 181 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √543 rounded to two decimal places?
√543 ≈ 23.30 to two decimal places (23.3 to one, 23.302 to three). Check: 23.30² = 542.89, close to 543.