Problem 48 - Algebra 1

Question

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$\frac{1 \pm 6 \sqrt{8}}{6}$$

Step-by-Step Solution

Answer

The rounded off values are 3.00 and -2.83

Step 1: Evaluate the square root

First, calculate the square root which is \(\sqrt{8}\). Use a calculator to find this, it is approximately 2.83

Step 2: Calculate the addition and subtraction

Next, calculate the expressions for both the plus and the minus. That is, replace the ± with + and with -. For \(1 + 6 \times 2.83\), the result is approximately 17.98. For \(1 - 6 \times 2.83\), the result is approximately -16.98.

Step 3: Perform the division

Now, divide each of these results by 6. Therefore, \(17.98 \div 6\) gives approximately 3.00 and \(-16.98 \div 6\) gives approximately -2.83.

Step 4: Round to the nearest hundredth

Round the results to the nearest hundredth to get final results. The result of \(3.00\) is already at the nearest hundredth, so it stays the same. However, \(-2.83\) is also already at the nearest hundredth, so it stays the same.

Key Concepts

Square Root Calculation

When calculating a square root, we search for a number which, when multiplied by itself, gives the original number. In the given expression, we have \( \sqrt{8} \). If you're unsure of square roots, remember that finding them typically requires a calculator, particularly for numbers not being perfect squares. Here, we use a calculator to determine that \( \sqrt{8} \) is approximately 2.83.
Understanding when and how to use this operation is crucial in algebra. It lets you simplify expressions and solve equations effectively.
A few vital points to remember about square roots:

• Integer results in square roots occur only for perfect squares such as 4, 9, 16, etc.
• If you're working by hand, approximate your results, especially with non-perfect squares.
• Using a calculator can speed up the process and improve accuracy.

Expression Simplification

Expression simplification involves breaking down a complex expression into more manageable parts. In the provided example, after determining \( \sqrt{8} \), we simplify the expression \( \frac{1 \pm 6 \times 2.83}{6} \).
Here we face two terms due to the \( \pm \) symbol, invoking both addition and subtraction, leading us to:

• \( 1 + 6 \times 2.83 = 17.98 \)
• \( 1 - 6 \times 2.83 = -16.98 \)

This step requires carefully following the mathematical operations order—multiplication before addition and subtraction. This fundamental skill allows you to break down barriers in mathematics by reducing larger problems into simpler, logical sequences.

Rounding Numbers

Rounding numbers is crucial, especially when dealing with decimal points. It helps present data in a more comprehendible manner and adjusts values for practical use. Following the expression simplification, we obtain approximate results: \( 17.98 \) and \( -16.98 \). When these numbers are divided by six, we achieve \( 3.00 \) and \( -2.83 \).
Now we focus on rounding to the nearest hundredth. Luckily, both results are already in this simplified form. However, understanding rounding rules is essential:

• Look at the thousandth place to determine if the hundredth place rounds up or stays the same.
• If the thousandth digit is 5 or more, round the hundredth digit up.
• If it is less than 5, keep the hundredth digit unchanged.

Grasping these rounding principles ensures precision in your answers, especially in scientific and engineering contexts where such operations are frequent.