Problem 11 - Intermediate Algebra

Question

Add or subtract. $$ \frac{5 \sqrt{2}}{3}+\frac{2 \sqrt{2}}{5} $$

Step-by-Step Solution

Answer

\(\frac{31 \sqrt{2}}{15}\)

Step 1: Identify the Denominator

The first step is to identify the denominators of the two fractions. Here, the denominators are 3 and 5.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator for the fractions with denominators 3 and 5 is 15.

Step 3: Convert Each Fraction to the Common Denominator

To convert \(\frac{5 \sqrt{2}}{3}\) to the common denominator 15, multiply both the numerator and denominator by 5: \(\frac{(5 \sqrt{2}) \times 5}{3 \times 5} = \frac{25 \sqrt{2}}{15}\). Convert \(\frac{2 \sqrt{2}}{5}\) to the common denominator 15 by multiplying both numerator and denominator by 3: \(\frac{(2 \sqrt{2}) \times 3}{5 \times 3} = \frac{6 \sqrt{2}}{15}\).

Step 4: Add the Fractions

Now that both fractions have the same denominator, add the numerators while keeping the denominator: \(\frac{25 \sqrt{2}}{15} + \frac{6 \sqrt{2}}{15} = \frac{(25 \sqrt{2} + 6 \sqrt{2})}{15}\).

Step 5: Simplify the Expression

Combine the terms in the numerator: \(25 \sqrt{2} + 6 \sqrt{2} = 31 \sqrt{2}\). Therefore, the expression simplifies to \(\frac{31 \sqrt{2}}{15}\).

Key Concepts

Understanding the Least Common Denominator (LCD)

To add or subtract fractions, having the same denominator is crucial. This common baseline is called the Least Common Denominator (LCD). It is the smallest number that can be divided evenly by all denominators in the problem.

Here's how to determine the LCD:

• List the multiples of each denominator. For example, for 3 the multiples are 3, 6, 9, 12, 15, and so on. For 5, they are 5, 10, 15, 20, and so forth.
• Find the smallest multiple that appears in both lists. That’s your LCD. In our example, 15 is the first common multiple.

Finding the LCD helps in aligning the fractions so they can be combined easily.
It prepares the fractions for effective addition or subtraction.

The Process of Fraction Addition

Once the fractions have the same denominators (thanks to the LCD), the addition process becomes straightforward. You only add the numerators while keeping the denominator unchanged.

Let’s see how it works:

• Take \(\frac{25 \sqrt{2}}{15}\) and \(\frac{6 \sqrt{2}}{15}\), which are already converted to the same denominator.
• To add them, simply add the numerators: \(25 \sqrt{2} + 6 \sqrt{2}\).
• The denominator stays the same, which is 15 in this case.

Now the expression looks like this: \(\frac{(25 \sqrt{2} + 6 \sqrt{2})}{15}\). By maintaining the same denominator, the fractions can combine smoothly, resulting in a new fraction.

Simplifying Fractions for the Final Answer

After adding or subtracting fractions, it's important to simplify the result. Simplifying makes fractions easier to understand and work with.

This is how to simplify:

• First, look at the numerator. If it comprises similar terms, combine them, like we have with \(25 \sqrt{2} + 6 \sqrt{2}\).
• This combination results in \(31 \sqrt{2}\) as the new numerator.

The final expression is \(\frac{31 \sqrt{2}}{15}\). Often, you’ll check if the numerator and denominator have common factors to reduce further, but in this case, they do not.

So, the fraction is simplified and ready as the final answer.