Problem 72 - Algebra 1

Question

Evaluate the expression to the nearest hundredth. $$ \frac{2 \pm 6 \sqrt{3}}{3} $$

Step-by-Step Solution

Answer

The two results are approximately 4.13 and -2.80

Step 1: Calculate positive square root

Calculate \(6 \sqrt{3}\) which means multiply 6 by the square root of 3. By using a calculator, you find that it equals approximately 10.39.

Step 2: Perform Addition and Subtraction Separately

Now, we perform the addition and subtraction operations separately, which gives us two results: (2 + 10.39) and (2 - 10.39). The result of these two operations will be approximately 12.39 and -8.39 respectively.

Step 3: Divide by 3

For the final step, divide both results of step 2 by 3. The first result (12.39) divided by 3 is approximately 4.13; the second result (-8.39) divided by 3 is approximately -2.80.

Key Concepts

Square Roots

Square roots are fundamental in mathematics, often representing a number which, when multiplied by itself, produces a given number. For example, the square root of 9 is 3 because 3 multiplied by itself gives 9.
When you encounter a square root, it's helpful to think of it as asking, “What number, times itself, produces this number?” This concept allows us to manage and simplify expressions that involve roots.
In the example expression, we have the term \(6 \sqrt{3}\). Here, \(\sqrt{3}\) is the number which, when squared, results in 3.
Multiplying it by 6 scales it up, and using a calculator, it evaluates to approximately 10.39. Understanding square roots helps in breaking down more complex terms in mathematical expressions, providing a clearer pathway to solving them.

Addition and Subtraction of Radicals

Radicals can seem complex, but simplifying them is a lot like working with regular numbers. If the 'radicand' (the number under the square root) is the same, we can add and subtract them just like regular numbers.
Take the simplified result from the square root calculation: \(10.39\). Here, we address the expression \(2 \pm 10.39\). This translates to two separate operations: addition and subtraction:

• \(2 + 10.39 = 12.39\)
• \(2 - 10.39 = -8.39\)

It’s important to evaluate both, as these will guide us to the next phase of the problem. Understanding how to handle addition and subtraction of radicals aids in dissecting expressions, contributing significantly to finding precise solutions.

Division of Real Numbers

Division among real numbers is a basic, yet crucial arithmetic operation. Remember, the real numbers include all the rational and irrational numbers that can be located on the number line.
Given our previous results from addition and subtraction, we now focus on dividing these by 3.

• First, take 12.39 divided by 3. This results in approximately 4.13.
• Second, we divide -8.39 by 3, yielding approximately -2.80.

Division helps scale numbers up or down and allows us to interpret relationships correctly within expressions. In expressions involving both radicals and divisions, such as the original exercise, division is the step that converts complex radical-involved results into more manageable forms. This is often the final simplification step, providing the final answer after prior additions and subtractions.