√564 = √(4×141) = 2√141 ≈ 2×11.87 = 23.74 - Decision Point
Simplifying √564: A Clear Guide to √(4×141) and Its Approximate Value
Simplifying √564: A Clear Guide to √(4×141) and Its Approximate Value
When dealing with square roots of composite numbers, simplification is key to understanding and comparing values accurately. One commonly encountered example is √564. While seemingly complex at first glance, this expression can be simplified using basic number properties, making it easier to work with both mathematically and in real-world applications.
Understanding √564: Breaking It Down
Understanding the Context
The number 564 isn’t a perfect square, but it’s a product of a perfect square and another integer:
√564 = √(4 × 141)
Since the square root of a product equals the product of the square roots, we apply the property:
√(a × b) = √a × √b
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Key Insights
So,
√(4 × 141) = √4 × √141
Because √4 = 2, we simplify further:
2 × √141
Now, √141 cannot be simplified further because 141 factors into 3 × 47, and neither factor is a perfect square. However, we can approximate its value for practical use.
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Approximating √141
Since √141 isn’t an exact integer, we estimate it using known square roots:
- We know that 11² = 121
- 12² = 144
Since 141 is between 121 and 144, √141 lies between 11 and 12.
A closer estimate:
Try 11.8² = 139.24
Try 11.9² = 141.61
So, 11.8² = 139.24 and 11.9² = 141.61 → 141 is closer to 141.61 than 139.24
Therefore, √141 ≈ 11.87 (accurate to two decimal places)
Final Calculation
Using this approximation:
2 × √141 ≈ 2 × 11.87 = 23.74
So,
√564 = 2√141 ≈ 23.74
