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

√564 = √(4×141) = 2√141 ≈ 2×11.87 = 23.74

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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Final Thoughts

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