Problem 13 - Algebra 1: Concepts and Skills

Question

Use a calculator or a table of square roots to evaluate the expression. Round the results to the nearest hundredth. $$ 6 \pm 5 \sqrt{3} $$

Step-by-Step Solution

Answer

The result of the expression \(6 + 5\sqrt{3}\) is approximately 14.66 and the result for \(6 - 5\sqrt{3}\) is approximately -2.66, both to the nearest hundredth.

Step 1: Calculation of Positive Scenario

To solve \(6+5\sqrt{3}\), first calculate the square root of 3. Using the calculator, you'll find \(\sqrt{3}\) is approximately 1.732. Then multiply this by 5 and add 6. These operations yield approximately 14.66 when rounded to the nearest hundredth.

Step 2: Calculation of Negative Scenario

Next, solve \(6 - 5\sqrt{3}\). Like the first scenario, first calculate 5 times the square root of 3, which is again approximately 8.66. This time, subtract this from 6, which gives a result of approximately -2.66 when rounded to the nearest hundredth.

Key Concepts

Calculator Use

When faced with the task of evaluating expressions that involve square roots, such as \(6 \pm 5\sqrt{3}\), a calculator becomes an invaluable tool. Calculators, whether physical or app-based, can accurately determine the square root of any number, which is critical when precise calculations are required. For instance, entering \(\sqrt{3}\) into the calculator yields approximately 1.732. This value can then be used in subsequent calculations to multiply by 5, as in our example exercise.

Modern calculators also allow you to store intermediate results, which can prevent errors during manual calculations and streamline the process. Besides, they often have functions for rounding numbers, which can be useful once your calculations are complete and you need to present a tidy answer.

Rounding Numbers

After using the calculator to perform necessary operations, results often need to be rounded to a specific decimal place. Rounding makes numbers more manageable and provides an answer that is precise enough for practical purposes. In our exercise, results are rounded to the nearest hundredth.

Let's consider the calculation for \(6 + 5\sqrt{3}\), which yields approximately 14.66073. According to rounding rules, if the digit in the thousandths place (third decimal) is 5 or more, increase the digit in the hundredths place (second decimal) by 1. Thus, the result becomes approximately 14.66.

Rounding is particularly important in mathematics for clarity and conciseness, especially when reporting results in a real-world context.

Positive and Negative Solutions

In mathematics, expressions like \(6 \pm 5\sqrt{3}\) demonstrate the idea of both positive and negative solutions. This expression can be split into two distinct parts: one where you add \(5\sqrt{3}\) to 6, and another where you subtract it.

In the **positive scenario** (\(6 + 5\sqrt{3}\)), you first find \(5\times\sqrt{3}\), which is approximately 8.66. Adding this to 6 gives us approximately 14.66, rounded.

Conversely, in the **negative scenario** (\(6 - 5\sqrt{3}\)), you still calculate \(5\times\sqrt{3}\) as approximately 8.66, but this time you subtract it from 6, resulting in roughly -2.66 when rounded.

Understanding positive and negative solutions is fundamental, especially when solving equations, as they can often reveal an entirely different set of answers to interpret and consider.