what is the square root of 567
The correct answer and explanation is:
The square root of 567 is approximately 23.811.
To understand this, first recall that the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25.
Finding the square root of 567 exactly can be challenging since 567 is not a perfect square (a perfect square is a number whose square root is an integer). Therefore, the square root of 567 is an irrational number, meaning it cannot be expressed exactly as a simple fraction or integer.
One way to estimate the square root of 567 is by using a method such as long division or a calculator. However, it also helps to look at the perfect squares around 567. For example:
- 23 squared is 529 (23 × 23 = 529)
- 24 squared is 576 (24 × 24 = 576)
Since 567 lies between 529 and 576, the square root must be between 23 and 24. Because 567 is closer to 576 than 529, the square root is closer to 24.
More precisely, using a calculator or numerical methods gives about 23.811. This means:
23.811 × 23.811 ≈ 567
Breaking down the number 567 into prime factors can also help simplify the square root expression. The prime factorization of 567 is:
567 = 3 × 3 × 3 × 3 × 7 = (3²) × (3²) × 7
Grouping the squares, we have:
√567 = √(3² × 3² × 7) = 3 × 3 × √7 = 9√7
Since √7 is approximately 2.6458, multiplying 9 by 2.6458 gives:
9 × 2.6458 ≈ 23.811
This matches the approximate decimal value found before.
In summary, the square root of 567 is an irrational number about 23.811, which can also be expressed as 9 times the square root of 7. This value, when squared, returns the original number, 567.