Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt{58}$$

Step-by-Step Solution

Answer

The decimal approximation of \(\sqrt{58}\) is 7.616.

Step 1: Understanding the Problem

We need to find the decimal approximation of the square root of 58 and round it to the nearest thousandth (three decimal places).

Step 2: Estimating the Square Root

Identify two integers between which the square root of 58 lies. Since 49 (7 squared) is less than 58 and 64 (8 squared) is more than 58, the square root of 58 is between 7 and 8.

Step 3: Calculation Using a Calculator

Use a calculator to find the value of \( \sqrt{58} \). Enter the value into the calculator to get an unrounded decimal. On most calculators, this would yield approximately 7.6157731.

Step 4: Rounding to the Nearest Thousandth

From the calculator's output, 7.6157731, identify the numeral in the thousandth place, which is 5. The digit following it is 7. Since 7 is 5 or more, round the thousandth place up by one.

Step 5: Final Answer

After rounding 7.6157731 to the nearest thousandth, the decimal approximation of \(\sqrt{58}\) is 7.616.

Key Concepts

Square Root

When we talk about the square root, we refer to a number which, when multiplied by itself, gets you the original number you're interested in. It's like asking which number times itself gives you 58. In mathematical terms, the square root is denoted by the symbol \(\sqrt{}\). So, when you see \(\sqrt{58}\), you're looking for the number that squared equals 58.

Finding square roots can be tricky, especially when the number isn't a perfect square—numbers like 1, 4, 9, etc., which have whole numbers as square roots. With numbers like 58, you'll often need to get an approximation. You start by knowing that since \(7^2 = 49\) and \(8^2 = 64\), \(\sqrt{58}\) has to be somewhere between 7 and 8.

Rounding

Rounding simplifies numbers by keeping them close to their original value but with fewer digits. When you round to the nearest thousandth, you're looking at the third digit after the decimal point. Why do we do this? Mainly, to make numbers easier to understand and work with in subsequent calculations.

Let's say our calculator gives us a number like 7.6157731 for \(\sqrt{58}\). The thousandth place here is 5, but we look at the next digit which is 7. Since 7 is more than five, the rule is to round up. This makes our final rounded number 7.616.

Calculator Use

Using a calculator is very helpful when working with tricky numbers like the square root of 58. Most scientific calculators have a square root function. Typically, you would:

• Turn on your calculator
• Locate the square root button (often denoted as \(\sqrt{}\) or "√")
• Input the number 58 and press the square root function
• Read the full decimal value given by the calculator

This gives you a precise, though often long, decimal number. For \(\sqrt{58}\), your calculator should display something close to 7.6157731.

Estimating Values

Estimating values is a handy skill when you need a rough idea without the need for exact numbers. Before using a calculator, you can estimate \(\sqrt{58}\) by recognizing it falls between square roots of perfect squares that are easy to calculate. We know

• \(\sqrt{49} = 7\)
• \(\sqrt{64} = 8\)

So \(\sqrt{58}\) should be somewhere between 7 and 8. A good initial estimate might be around 7.5 or 7.6. It's useful for checking the plausibility of your eventual calculated answer.