Problem 16 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{5}{m}+\frac{4}{m}}{\frac{5}{m}-\frac{4}{m}} $$

Step-by-Step Solution

Answer

Question: Simplify the complex rational expression: $$\frac{\frac{5}{m}+\frac{4}{m}}{\frac{5}{m}-\frac{4}{m}}$$ Solution: The simplified complex rational expression is $$9$$.

Step 1: Find a common denominator for the numerator and denominator fractions

As we can see, both fractions in the numerator have the same denominator, m, and both fractions in the denominator also have the same denominator, m. Since they already have the same denominator, we can move to the next step.

Step 2: Combine the fractions in the numerator and denominator

Now, we can add the fractions in the numerator: $$ \frac{5}{m} + \frac{4}{m} = \frac{5+4}{m} = \frac{9}{m} $$ Then, let's subtract the fractions in the denominator: $$ \frac{5}{m} - \frac{4}{m} = \frac{5-4}{m} = \frac{1}{m} $$ So now, the expression looks like this: $$ \frac{\frac{9}{m}}{\frac{1}{m}} $$

Step 3: Simplify the complex rational expression

Finally, we have a fraction divided by another fraction. In general, dividing by a fraction is the same as multiplying by its reciprocal. So we will multiply the numerator by the reciprocal of the denominator: $$ \frac{9}{m} \times \frac{m}{1} $$ Next, we can cancel the m terms in the numerator and denominator: $$ \frac{9 \cancel{m}}{\cancel{m}} \times \frac{\cancel{m}}{1} = \frac{9}{1} $$ And thus, the simplified complex rational expression is: $$ \boxed{9} $$

Key Concepts

Finding Common Denominators

When working with complex rational expressions, the first important step is finding a common denominator. This ensures that you can easily add or subtract fractions. A denominator is the bottom part of a fraction, and when it's consistent across different terms, combining them becomes straightforward.

In our exercise, both the numerator and the denominator parts of the complex fraction have fractions with the same denominator, \(m\). This is beneficial as it simplifies our work; there is no need to find a least common denominator when they are already the same. With shared denominators, it's easy to

• Add fractions in the numerator or denominator.
• Subtract fractions as well.

By recognizing a common denominator early, the operation steps become much smoother.

Fraction Operations

Fraction operations are essential in simplifying complex rational expressions. They involve adding, subtracting, multiplying, and sometimes dividing fractions. Each operation requires particular rules.

In the exercise, we see fractions like \(\frac{5}{m}\) and \(\frac{4}{m}\). Since they share a denominator, addition and subtraction are straightforward:

• Add numerators directly while keeping the common denominator; thus, \(\frac{5}{m} + \frac{4}{m} = \frac{9}{m}\).
• Subtract numerators similarly; hence, \(\frac{5}{m} - \frac{4}{m} = \frac{1}{m}\).

After simplifying numerator and denominator separately, the complex fraction reduces to \(\frac{\frac{9}{m}}{\frac{1}{m}}\). This step completes the integration of fractions in each part of the main fraction.

Simplification Steps

Simplification steps transform a complex expression into a simpler form. This often involves reducing fractions or cancelling out terms. The main goal is achieving a clean, final expression without unnecessary complexity.

For our complex rational expression, the final step involves simplifying \(\frac{\frac{9}{m}}{\frac{1}{m}}\). Think of it as dividing by a fraction, which is equivalent to multiplying by its reciprocal. So:

• Swap division for multiplication with the reciprocal: \(\frac{9}{m} \cdot \frac{m}{1}\).
• Cancel the \(m\) terms since they appear both above and below the fraction line.

This simplifies everything down to a simple whole number: 9. Thus, transforming complex mathematical statements into something much more digestible.