√528 at a glance
- Exact value
- 4√33
- Decimal (10 places)
- 22.9782505862
- Rounded
- 23.0 · 22.98 · 22.978
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.978251
- Prime factorization
- 2⁴ × 3 × 11
- Cube root
- 8.082480
How to simplify √528
Look for the largest perfect square that divides 528. Here it is 16 (4²), because 528 = 16 × 33 and 33 has no square factor left:
The prime factorization tells the same story: 528 = 2⁴ × 3 × 11. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 3 × 11 stays inside.
528 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √528 = 2√132, and √132 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√33)² = 4² × 33 = 16 × 33 = 528. As a decimal, 4√33 = 4 × 5.7445626465 ≈ 22.9782505862.
Where √528 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √528 lies between 22 and 23. 528 is 44 above 484 and 1 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.9778 (0% low)
- Tangent from 22, i.e. 22 + 44 ÷ 44: 23.0000 (0.09% high)
- Tangent from 23, i.e. 23 − 1 ÷ 46: 22.9783 (0% high)
For √528 the tangent at 23 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 528 is just 1 below 529.
Finding √528 with the Babylonian method
Picture a rectangle with an area of 528 and one side x; the other side must be 528 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √528.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 528 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.9565217391 | 22.9782608696 | 4 |
| 2 | 22.9782608696 | 22.9782403027 | 22.9782505862 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √528 = 22.9782505862 to every decimal shown.
√528 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √528 the pattern is [22; 1, 44] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √528 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 9.8 × 10⁻¹ |
| 23/1 | 23.0000000000 | 2.2 × 10⁻² |
| 1,034/45 | 22.9777777778 | 4.7 × 10⁻⁴ |
| 1,057/46 | 22.9782608696 | 1.0 × 10⁻⁵ |
| 47,542/2,069 | 22.9782503625 | 2.2 × 10⁻⁷ |
| 48,599/2,115 | 22.9782505910 | 4.9 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 528y² = 1. Its smallest solution in positive whole numbers is x = 23, y = 1.
√528 in geometry and everyday measurements
- A square garage floor of 528 square feet measures about 22.98 ft (23 ft) per side, and its corner-to-corner diagonal is √1056 ≈ 32.5 ft.
- 528 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √528 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 16 × 16 box, because 4² + 16² + 16² = 528.
- Since √528 = 4√33, a length of √528 is exactly 4 copies of the length √33 laid end to end.
Square roots near √528 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √525 | 5√21 | 22.9129 | No |
| √526 | √526 | 22.9347 | No |
| √527 | √527 | 22.9565 | No |
| √528 | 4√33 | 22.9783 | No |
| √529 | 23 | 23.0000 | Yes |
| √530 | √530 | 23.0217 | No |
| √531 | 3√59 | 23.0434 | No |
- The cube root of 528 is about 8.082480.
- Because 528 = 4 × 132, the root is twice √132: 2 × 11.489125 ≈ 22.978251.
Frequently asked questions
What is the square root of 528?
The square root of 528 is 4√33 in simplest radical form, which is about 22.9782505862. The negative root, −22.978251, also squares to 528.
Is the square root of 528 rational or irrational?
Irrational. 528 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √528 be simplified?
Yes. The largest perfect square dividing 528 is 16, so √528 = √16 × √33 = 4√33.
What is √528 rounded to two decimal places?
√528 ≈ 22.98 to two decimal places (23.0 to one, 22.978 to three). Check: 22.98² = 528.0804, close to 528.