/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 \(\sqrt{23.2}\)... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 73

\(\sqrt{23.2}\)

Short Answer

Expert verified
The square root of 23.2 is approximately 4.8.

Step by step solution

01

Estimate the Square Root

First, find two perfect squares between which 23.2 falls. Notice that 23.2 is between 16 and 25, and their square roots are 4 and 5 respectively. Therefore, \( \sqrt{23.2} \) should be between 4 and 5.
02

Refine the Estimate

Try a number between 4 and 5 to refine the estimate. Since \( \sqrt{23.2} \approx 4.8 \), we can try calculating \( 4.8^2 \). We get \( 4.8 \times 4.8 = 23.04 \), which is close to 23.2.
03

Verify the Answer

Finally, we can compare the result of \( 4.8^2 \) to 23.2. Since 23.04 is very close to 23.2, our estimation \( \sqrt{23.2} \approx 4.8 \) is reasonable.

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Key Concepts

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

Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9. When we estimate square roots, we want to find a number that closely approximates the square root of a given number. This process is essential when dealing with non-perfect squares, such as 23.2.
Perfect Squares
A perfect square is a number that is the square of an integer. Examples of perfect squares include 1, 4, 9, 16, 25, and so on. They are important in mathematical estimation because they act as reference points. For example, in estimating the square root of 23.2, we look for the perfect squares surrounding 23.2, which are 16 and 25. These help us narrow down where the square root of 23.2 falls, since \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \), we know \( \sqrt{23.2} \) must be between 4 and 5.
Mathematical Estimation
Mathematical estimation is a technique used to find an approximate value when an exact answer is not necessary or difficult to calculate. Estimating square roots involves comparing the number to nearby perfect squares and choosing a value that is between those perfect squares. For instance, we estimated \( \sqrt{23.2} \) by first identifying that it is between the square roots of 16 and 25 (which are 4 and 5). We further refined the estimate by selecting a midpoint, such as 4.8, and squaring it to see if it was close to 23.2. Verifying the estimate ensures that our approximation is reasonable.

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