Problem 76 - College Algebra with Modeling and Visualization

Question

Use a calculator to evaluate the expression. Round your result to the nearest thousandth. $$ \frac{5+\sqrt{5}}{2} $$

Step-by-Step Solution

Answer

The expression evaluates to approximately 3.618.

Step 1: Calculate the Square Root

First, find the square root of 5. This is approximately \( \sqrt{5} \approx 2.236 \).

Step 2: Perform Addition

Add the square root of 5 to 5. So, you have: \( 5 + \sqrt{5} \approx 5 + 2.236 = 7.236 \).

Step 3: Perform Division

Divide the result by 2: \( \frac{7.236}{2} \approx 3.618 \).

Step 4: Round the Result

Finally, round the result to the nearest thousandth. Hence, \( 3.618 \) remains unchanged, as it is already to the nearest thousandth.

Key Concepts

Square Roots

The concept of a square root is fundamental in algebra. When we talk about the square root of a number, we are referring to a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because multiplying 5 by itself (5×5) equals 25. This is represented mathematically as \( \ \sqrt{25} = 5 \ \).
When calculating square roots of non-perfect squares, the values are irrational, meaning they can't be expressed as a simple fraction. They are often left in square root form or approximated as decimals. For example, \( \ \sqrt{5} \ \) is approximately 2.236.
Calculating a square root manually can be complex, but calculators can quickly provide the decimal approximation. When solving expressions involving square roots, it's essential to calculate this part accurately to ensure the correct final result.

Addition and Division

After finding the square root of a number, the next steps in our problem often involve addition and division operations. Let's break these down:

• Addition: This is one of the simplest arithmetic operations where you combine two numbers to get a sum. In our exercise, once the square root of 5 is determined, we add it to the number 5. The process is straightforward: you simply align the digits and perform the addition. For example, \( \ 5 + 2.236 = 7.236 \ \).
• Division: Dividing a number means splitting it into equal parts. When we divide 7.236 by 2, we are essentially finding out how many times 2 fits into the number 7.236. This operation yields \( \ \frac{7.236}{2} = 3.618 \ \). Division requires accuracy especially with decimal numbers to ensure the preciseness of the result.

Understanding these operations is crucial as small errors in addition or division can lead to incorrect results. Therefore, it's always good practice to double-check calculations, especially when dealing with decimals.

Rounding Numbers

Rounding numbers is a valuable skill, especially when working with decimals. It allows us to simplify numbers while maintaining their approximate value. In mathematical terms, rounding to the nearest thousandth means looking at the fourth digit after the decimal point. If this digit is 5 or more, you round up. If it's less than 5, you keep the number as is.
In our example, after performing all calculations, we arrive at 3.618. This number is already rounded to the nearest thousandth because the fourth digit (following the hundredth place) is less than 5. As a result, there's no need for adjustment.
Rounding becomes very handy when dealing with long decimal results, as it helps express answers clearly without a lengthy string of numbers. It is crucial in ensuring your answer is both precise and easy to interpret, especially in academic settings.