Problem 62 - Prealgebra

Question

Estimate each square root to the nearest whole number. Do not use a calculator. (Lesson \(9-1\) ) $$\sqrt{54}$$

Step-by-Step Solution

Answer

The square root of 54 is approximately 7.

Step 1: Identify Perfect Squares

To estimate the square root of 54, first identify the two perfect squares between which this number falls. The perfect squares around 54 are 49 (since \(7^2 = 49\)) and 64 (since \(8^2 = 64\)).

Step 2: Write the Inequality

Establish that the square root of 54 lies between the square roots of 49 and 64. Thus, we have \(7^2 < 54 < 8^2\), which means \(7 < \sqrt{54} < 8\).

Step 3: Estimation

Since 54 is closer to 49 than it is to 64, \( \sqrt{54} \) is closer to 7 than it is to 8. Consider the difference from both endpoints: \(54 - 49 = 5\) and \(64 - 54 = 10\). Since 5 is less than 10, \( \sqrt{54} \) is closer to 7.

Key Concepts

Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself. For example, the number 9 is a perfect square because it is the result of multiplying 3 by itself (\(3^2 = 9\)). These numbers are essential when estimating square roots, as they act as benchmarks to determine where a non-perfect square falls between two known perfect squares.

Understanding perfect squares can simplify many mathematical problems. They are especially useful in estimating square roots without a calculator. In the given problem, 49 (\(7^2\)) and 64 (\(8^2\)) are perfect squares used to estimate \(\sqrt{54}\). By identifying that 54 falls between these two perfect squares, it provides a useful starting point for making a close approximation.

Inequality

Inequality is a mathematical concept where we compare the size of two values or expressions. When estimating square roots, inequality helps us place our target number between two more easily understandable numbers.

In the case of \(\sqrt{54}\), using inequality allows us to know that \(7^2 < 54 < 8^2\), meaning the actual square root of 54 is between 7 and 8. This is critical because this mathematical statement narrows down our possible values for \(\sqrt{54}\), giving us a close range to work within for further estimation. It's a tool that helps us confidently say that, although \(\sqrt{54}\) is not exactly 7 or 8, it must be more than 7 but less than 8.

Mathematical Estimation

Mathematical estimation involves making an educated guess about the exact value of a number. It's a practical approach when an exact solution is not necessary, or exact calculation is not feasible. In estimating square roots, we rely on the nearby perfect squares and inequalities to guide our guess.

For \(\sqrt{54}\), after determining the range (between 7 and 8), we refine our estimation by checking how close 54 is to these perfect squares. Calculating the differences, \(54 - 49 = 5\) and \(64 - 54 = 10\), we observe that 54 is 5 units away from 49, and 10 units away from 64. This indicates that \(\sqrt{54}\) is closer to 7. Through this process, we conclude that the best whole number estimate for \(\sqrt{54}\) is 7, since it seems closer to this end of the range.