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Chapter 9: Problem 22

Estimate each square root between two consecutive whole numbers. $$ \sqrt{55} $$

Short Answer

Expert verified
\( \sqrt{55} \) is between 7 and 8.

Step by step solution

01

- Identify perfect squares around the number

To estimate \( \sqrt{55} \), first identify the two closest perfect squares around 55. The perfect square less than 55 is 49 (since \( \sqrt{49} = 7 \)) and the perfect square greater than 55 is 64 (since \( \sqrt{64} = 8 \)).
02

- Determine the consecutive whole numbers

Since 49 is less than 55 and 64 is more than 55, \( \sqrt{55} \) lies between the square roots of these numbers. Therefore, \( \sqrt{55} \) is between 7 and 8.
03

- Estimate closer

To get a better estimate, note that 55 is closer to 49 than to 64. Hence, \( \sqrt{55} \) is nearer to 7 than to 8. One can further refine this, but for the purpose of this exercise, determining it lies between 7 and 8 is sufficient.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

perfect squares
A perfect square is a number that is the result of squaring an integer. This means multiplying an integer by itself.
For example, 4 is a perfect square because it is the result of squaring 2 (i.e., 2 脳 2 = 4). Similarly, 9 is a perfect square because it's the result of squaring 3 (i.e., 3 脳 3 = 9).

To identify perfect squares around a given number, you can find the square roots of perfect squares close to that number. In the case of estimating \(\sqrt{55}\), the perfect squares closest to 55 are 49 (since \(\sqrt{49}\) = 7) and 64 (since \(\sqrt{64}\) = 8). By knowing the perfect squares, we can narrow down our estimate for the square root.
consecutive whole numbers
Consecutive whole numbers follow each other in order without any gaps. When estimating a square root, finding two consecutive whole numbers between which the square root lies helps us zero in on the approximation.

For instance, recognizing that the perfect squares around 55 are 49 and 64 means \(\sqrt{55}\) lies between 7 and 8 (since \(\sqrt{49}\) = 7 and \(\sqrt{64}\) = 8). This tells us that \(\sqrt{55}\) is between these two consecutive whole numbers. It is a vital step in estimating square roots accurately.
approximation of square roots
The final step in estimating a square root is to make a closer approximation. Once we know the two consecutive whole numbers between which the square root lies, we can further refine our estimate.

For \(\sqrt{55}\), since 55 is closer to 49 than to 64, we can infer that \(\sqrt{55}\) is nearer to 7. To make an even closer approximation, consider the distance between the two consecutive perfect squares:
  • 55 is 6 units away from 49.
  • 55 is 9 units away from 64.
This means \(\sqrt{55}\) is between 7 and 8 but closer to 7. While this exercise might not need such precision, having this understanding helps in refining our guesses in other scenarios.

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