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Chapter 5: Problem 712

In the following exercises, simplify. Estimate \(\sqrt{54}\) to between two whole numbers.

Short Answer

Expert verified
The value of \(\sqrt{54}\) is between 7 and 8.

Step by step solution

01

Determine the perfect squares around 54

Identify the two closest perfect squares on either side of 54. These perfect squares are 49 (since \(7^2 = 49\)) and 64 (since \(8^2 = 64\)).
02

Determine the integers

Since \(7^2 = 49\) and \(8^2 = 64\), this indicates that \(\sqrt{54}\) lies between 7 and 8.

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Key Concepts

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

perfect squares
Perfect squares are numbers that are the square of an integer. In other words, when you multiply a number by itself, the result is a perfect square. For example,
  • 3 multiplied by 3 equals 9, so 9 is a perfect square.
  • 4 multiplied by 4 equals 16, so 16 is a perfect square.
To estimate the square root of a number, it is key to know the perfect squares close to it. These reference points help to narrow down the possible values.
In the exercise, we need to estimate \(\sqrt{54}\). The perfect squares closest to 54 are:
  • 49, since \(\7^2 = 49\).
  • 64, since \(\8^2 = 64\).
Understanding perfect squares makes the process of estimating square roots much easier.
integer estimation
Integer estimation involves finding two whole numbers between which a given number lies. For square roots, this means identifying which two perfect squares the original number is between.
In our exercise, we know that \(\sqrt{54}\) lies between the perfect squares 49 and 64.
This tells us that the integer part of \(\sqrt{54}\) is between 7 and 8 because:
  • 7 is the square root of 49.
  • 8 is the square root of 64.
So, we can confidently say that \(\sqrt{54}\) is somewhere between 7 and 8. Simple, right? Just remember to identify the closest perfect squares, then find the integers connected to them.
square root simplification
Square root simplification is the process of finding the closest and most accurate value of a square root. Once you have estimated the square root to lie between two integers, the next step is to refine that estimate. While our exercise stops at the integer estimation, here's how you could proceed to simplify further:
  • Determine if the square root is closer to the lower or higher number: Since 54 is closer to 49 than to 64, we can guess that \(\sqrt{54}\) is closer to 7.5 than to 7 or 8.
  • Use decimal points for more precision: You might guess halfway points, like 7.4, 7.6, 7.8, etc. Then square these to see which is closest to 54.
For our case, \(\7.3^2 \approx 53.29\), and \(\7.4^2 \approx 54.76\). So, \sqrt{54}\ is a bit closer to 7.3 or 7.4. However, for many purposes, knowing it is between 7 and 8 is detailed enough. Square root simplification enhances understanding and improves accuracy. Keep practicing and it will become second nature!

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