Problem 22 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{a}{4}+\frac{7 a}{8}}{\frac{6 a^{2}}{5}-\frac{3 a^{2}}{10}}+\frac{3}{a}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{17}{4a}\), undefined at \(a = 0\).

Step 1: Simplifying the Numerator

We start by simplifying the numerator of the given fraction. The numerator is \(\frac{a}{4} + \frac{7a}{8}\). Find the least common denominator (LCD), which is 8. Rewrite each fraction with the LCD: \(\frac{2a}{8}\) for \(\frac{a}{4}\) and keep \(\frac{7a}{8}\) as is. So, the numerator becomes \(\frac{2a + 7a}{8} = \frac{9a}{8}\).

Step 2: Simplifying the Denominator

Next, simplify the denominator \(\frac{6a^2}{5} - \frac{3a^2}{10}\). The LCD here is 10. Rewrite each fraction with the LCD: \(\frac{12a^2}{10}\) for \(\frac{6a^2}{5}\) and keep \(\frac{3a^2}{10}\) consistent. Thus, the denominator becomes \(\frac{12a^2 - 3a^2}{10} = \frac{9a^2}{10}\).

Step 3: Simplifying the Complex Fraction

Now, use the simplified numerator and denominator to form a new fraction: \(\frac{\frac{9a}{8}}{\frac{9a^2}{10}}\). Simplify by multiplying the numerator by the reciprocal of the denominator: \(\frac{9a}{8} \times \frac{10}{9a^2} = \frac{90a}{72a^2}\). Cancel the common terms and simplify to get \(\frac{5}{4a}\).

Step 4: Adding \(\frac{3}{a}\) to the Simplified Expression

Add \(\frac{3}{a}\) to the simplified expression \(\frac{5}{4a}\). The common denominator between \(4a\) and \(a\) is \(4a\). Rewrite \(\frac{3}{a}\) as \(\frac{12}{4a}\) to match the denominator. Combine to get \(\frac{5}{4a} + \frac{12}{4a} = \frac{17}{4a}\).

Step 5: Determine Values Where the Expression is Undefined

The fraction \(\frac{17}{4a}\) is undefined when the denominator is 0. Set the denominator to zero and solve for \(a\): \[4a = 0\]\[a = 0\]Thus, the expression is undefined for \(a = 0\).

Key Concepts

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form. When simplifying, you aim to make the numbers as small as possible without changing the fraction's value.
The process involves:

• Finding the greatest common factor (GCF) of the numerator and the denominator.
• Dividing both the numerator and the denominator by their GCF.

This results in an equivalent fraction that is simpler and easier to work with. In the earlier step, \( rac{9a}{8} \) and \( rac{9a^2}{10} \) were simplified by using the concept of reciprocals and canceling common terms. Remember, the goal is to make math operations straightforward and error-free.

Complex Fractions

Complex fractions appear intimidating because they contain fractions within fractions. However, simplifying them isn't as daunting once you get the hang of it.
A complex fraction like \(rac{rac{9a}{8}}{rac{9a^2}{10}}\) requires a few steps:

• Simplify the fractions in the numerator and the denominator separately if possible.
• Multiply by the reciprocal of the denominator fraction. This means flipping the denominator fraction and multiplying it with the numerator.
• Look to simplify the resulting fractions by canceling out common terms or simplifying further if needed.

After mastering this process, complex fractions become much simpler to navigate.

Common Denominators

To combine fractions efficiently, having a common denominator is key. A common denominator is a shared multiple of the denominators you're working with.
When adding or subtracting fractions, it's crucial for them to share this common denominator. Here's how you do it:

• Find the least common multiple (LCM) of the denominators.
• Adjust each fraction so they both have this LCM as their new denominator.

For example, in the expression \(rac{5}{4a}\) + \(rac{3}{a}\), the common denominator is \4a\. Converting \(rac{3}{a}\) to a fraction with a denominator \4a\ makes it easier to add to \(rac{5}{4a}\).

Undefined Expressions

Undefined expressions in fractions occur when the denominator equals zero. Division by zero is undefined in mathematics because it does not result in a meaningful value.
To find where a fraction is undefined:

• Set the denominator equal to zero and solve for the variable.

For instance, the expression \(rac{17}{4a}\) is undefined when \(4a = 0\). Solving this gives \(a = 0\), indicating \(a\) cannot be zero. Always check for this condition to avoid dealing with undefined or unreliable results, which can skew your calculations.