Problem 44 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{5}{7 t}+\frac{3}{4 t^{2}}+\frac{1}{14 t} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{22t + 21}{28t^2}\).

Step 1: Identify the Denominator

The given rational expressions have different denominators: \(7t\), \(4t^2\), and \(14t\). To perform the addition, we need to find a common denominator for these rational expressions.

Step 2: Determine the Least Common Denominator (LCD)

The least common denominator for \(7t\), \(4t^2\), and \(14t\) is \(28t^2\). This is the smallest expression that each of the denominators can divide into without a remainder.

Step 3: Rewrite Each Expression with the LCD

Convert each term to have the denominator \(28t^2\):- \(\frac{5}{7t} = \frac{5 \times 4t}{28t^2} = \frac{20t}{28t^2}\)- \(\frac{3}{4t^2} = \frac{3 \times 7}{28t^2} = \frac{21}{28t^2}\)- \(\frac{1}{14t} = \frac{1 \times 2t}{28t^2} = \frac{2t}{28t^2}\)

Step 4: Add the Numerators

Now add the numerators of the expressions over the common denominator:\(\frac{20t}{28t^2} + \frac{21}{28t^2} + \frac{2t}{28t^2} = \frac{20t + 21 + 2t}{28t^2}\)

Step 5: Combine Like Terms

Combine like terms in the numerator:\(20t + 2t + 21 = 22t + 21\).The expression is now \(\frac{22t + 21}{28t^2}\).

Step 6: Simplify the Expression

Check if the expression \(\frac{22t + 21}{28t^2}\) can be further simplified. As the numerator and denominator have no common factors, this expression is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When dealing with rational expressions, one common task is to add or subtract them. To do this efficiently, we focus on finding the least common denominator (LCD). The LCD is the smallest multiple that can be evenly divided by each of the given denominators. This allows us to rewrite each rational expression so they share a common denominator, making addition or subtraction straightforward.

In the example, the rational expressions have the denominators: \(7t\), \(4t^2\), and \(14t\). To find their LCD, we look for the smallest expression that each denominator can divide into without a remainder. Here, it is \(28t^2\).

Finding an LCD involves:

• Identifying the largest factors of each denominator.
• Combining those factors to form the smallest common multiple.

This step is crucial because it ensures the expressions are compatible for addition or subtraction.

The Art of Combining Like Terms

Once we have a common denominator, the next step is to focus on the numerators. Imagine having different pieces of a puzzle that need to form one image. The key is to combine like terms.

Combining like terms means grouping together similar parts to simplify the expression. In our example, after converting each term to have the denominator \(28t^2\), the expression becomes \(\frac{20t}{28t^2} + \frac{21}{28t^2} + \frac{2t}{28t^2}\).

You can see that:

• \(20t\) and \(2t\) are like terms because they both have the variable \(t\).
• Adding them together gives \(22t\).

The constant term \(21\) has no variable and is therefore left on its own. Accordingly, the numerator after combining like terms is \(22t + 21\). This process simplifies the equation step by step.

Simplifying Fractions to Their Finest

The final step in manipulating rational expressions is simplification. Simplifying fractions involves reducing them to their smallest possible form without altering their value. For our expression \(\frac{22t + 21}{28t^2}\), we check if there are any common factors to cancel both in the numerator and the denominator.

This fraction is already as simple as it can be because:

• The numerator \(22t + 21\) and the denominator \(28t^2\) don't share any common factors.
• Thus, there's no need for further reduction.

Simplifying fractions ensures clarity and simplicity in mathematical expressions, especially in more complex equations. It's like refining a diamond to reveal its true brilliance. Always remember, simplifying is about looking for commonalities that can tidy up the expression into a neat package.