Problem 54 - Prealgebra

Question

Use a calculator to work. Approximate each of the following expressions to the nearest thousandth. $$\sqrt{18}+\sqrt{50}$$

Step-by-Step Solution

Answer

11.314

Step 1: Calculate \( \sqrt{18} \)

Use your calculator to find the square root of 18. Type 18, then press the square root button. You should find that \( \sqrt{18} \approx 4.2426 \). Round this to the nearest thousandth to get \( 4.243 \).

Step 2: Calculate \( \sqrt{50} \)

Similarly, use your calculator to determine the square root of 50. Enter 50 and then press the square root button. You should see \( \sqrt{50} \approx 7.0711 \). Round this to the nearest thousandth to obtain \( 7.071 \).

Step 3: Add the Calculated Square Roots

Add the results from Step 1 and Step 2. This means you sum \( 4.243 \) and \( 7.071 \). \( 4.243 + 7.071 = 11.314 \). Round this to the nearest thousandth if needed (in this case, the rounding maintains 11.314).

Key Concepts

Understanding Approximations

Approximations are a vital concept in mathematics, especially when exact values are complex or impossible to obtain easily. Take square roots as an example, the exact square root of numbers like 18 and 50 are irrational numbers that continue infinitely without repeating. Approximations help us to use these numbers practically by rounding them to a specific decimal place. This practice is not only used in academic calculations, but also in real-world scenarios such as engineering and finance.

When you approximate, you choose to represent the number within a certain level of precision. In our exercise, we approximate to the nearest thousandth. A few reasons to approximate include:

• Simplifying calculations for quicker problem-solving.
• Making irrational numbers more manageable and easier to report.
• Maintaining clarity in communication by not having to write down heavily cluttered numbers.

These rules of approximation ensure our calculations remain simple, yet accurate enough for everyday tasks. It's about finding a balance between precision and practicality.

How to Use a Calculator for Square Roots

Calculators are powerful tools that greatly ease the process of finding square roots, especially for irrational numbers where manual calculation would be labor-intensive. Knowing how to correctly use a calculator ensures accuracy in your results. Here’s a quick guide on how to use calculators to find square roots:

• Ensure your calculator is in the right mode, typically standard or scientific mode, depending on its design.
• Enter the number you need the square root for. In our case, for calculating \( \sqrt{18} \) and \( \sqrt{50} \), you would type '18' or '50'.
• Locate and press the square root function button, usually marked as '√'. Some calculators might require you to press 'shift' or '2nd' before pressing the square root button.
• Read and note down the displayed number, which is the square root approximation given by your calculator.

Calculators take out the margin for manual error and allow quick repetition for practice or verification, significantly aiding in confidence and efficiency when handling square roots.

Mastering Mathematics Rounding

Rounding is a key mathematical skill that lets you refine your approximations by adjusting a number to the closest desired precision. When you round a number to the nearest thousandth, you look at the fourth decimal place to determine your action.

Here's a simple method to round to the nearest thousandth:

• Identify the thousandth place, which is the third digit to the right of the decimal point.
• Look at the digit immediately to the right (the fourth digit) to decide whether to round up or remain the same. If this digit is 5 or higher, round the thousandth place up by one. If it's 4 or lower, keep the thousandth place unchanged.
• Apply the standard rules of rounding consistently in all calculations for accuracy and consistency. In our exercise, \( \sqrt{18} \approx 4.2426 \) rounds to \( 4.243 \) and \( \sqrt{50} \approx 7.0711 \) rounds to \( 7.071 \).

Combining approximation with rounding not only ensures accuracy but also maintains the integrity and communicate correctness of results, which is essential in both academic and professional environments.