Problem 114 - Intermediate Algebra

Question

Add the rational expressions by expressing them in terms of a common denominator \(24 b^{3}\). (Note: This is not the LCD.) An extra step has to be performed to obtain the correct result because the lowest common denominator was not used. What was the step? $$\frac{r}{4 b^{2}}+\frac{s}{6 b}$$

Step-by-Step Solution

Answer

Add as \(\frac{6rb + 4sb^2}{24b^3}\) and simplify if possible.

Step 1: Identify the Denominators

The given rational expressions are \( \frac{r}{4b^2} \) and \( \frac{s}{6b} \). The denominators are \(4b^2\) and \(6b\). Your task is to express both of these fractions in terms of a common denominator, which is provided as \(24b^3\).

Step 2: Determine Multiplication Factors

To convert each fraction to have the denominator \(24b^3\), determine what each original denominator needs to be multiplied by. For \(4b^2\), multiply by \(\frac{6b}{6b}\) to get \(24b^3\). For \(6b\), multiply by \(\frac{4b^2}{4b^2}\) to get \(24b^3\). These fractions ensure the numerators are scaled appropriately for the new denominator.

Step 3: Rewrite Expressions with Common Denominator

Multiply the numerators by their respective factors determined in Step 2: \[ \frac{r \times 6b}{4b^2 \times 6b} = \frac{6rb}{24b^3} \]\[ \frac{s \times 4b^2}{6b \times 4b^2} = \frac{4sb^2}{24b^3} \] Thus, the two expressions are rewritten as \(\frac{6rb}{24b^3}\) and \(\frac{4sb^2}{24b^3}\).

Step 4: Add the New Fractions

Now that the fractions share the same denominator, add the numerators:\[\frac{6rb + 4sb^2}{24b^3}\]The operation combines both rational expressions under a single denominator.

Step 5: Simplify the Result

Even though the combined expression is over a common denominator of \(24b^3\), examine the numerator for further simplification. Since the common denominator \(24b^3\) was not the lowest possible common denominator, check if any terms in the numerator can be factored or canceled before concluding the simplification.

Step 6: Additional Step: Realize the Simplification

Because the chosen denominator isn't the lowest common denominator (L.C.D.), further simplification may not be necessary due to minimal simplification; however, always check whether the expression can be reduced by a common factor in both the numerator and denominator for optimal simplicity.

Key Concepts

Common Denominator

When working with rational expressions, especially those involving fractions, it’s crucial to find a common denominator to combine them effectively. A common denominator is essentially the same bottom number shared by all fractions in your problem. This allows you to add or subtract the fractions easily, as their denominators are now consistent.

In the case of the fractions \( \frac{r}{4b^2} \) and \( \frac{s}{6b} \), the exercise provides a common denominator of \( 24b^3 \).

To make the denominators match \( 24b^3 \):

• Multiply \( 4b^2 \) by \( \frac{6b}{6b} \) to get \( 24b^3 \).
• Multiply \( 6b \) by \( \frac{4b^2}{4b^2} \) to get \( 24b^3 \).

Now both fractions can be rewritten with this shared denominator, allowing for straightforward operations like addition.

Adding Fractions

Once rational expressions have a common denominator, adding them is straightforward. Simply add the numerators together while keeping the common denominator as it is.

In our exercise, the two expressions become \( \frac{6rb}{24b^3} \) and \( \frac{4sb^2}{24b^3} \) after finding the common denominator.

Add the numerators:

• Combine \( 6rb \) and \( 4sb^2 \) to get \( 6rb + 4sb^2 \).

This results in the expression: \[\frac{6rb + 4sb^2}{24b^3}\] Ensure that you keep the denominator \( 24b^3 \) the same while performing these operations. This habit ensures you will reach the correct answer consistently.

Remember, the key to adding fractions is all about having that common denominator!

Simplification of Fractions

Simplifying is typically the final step in working with fractions or rational expressions. It involves reducing the expression to its simplest form, which often means finding and canceling common factors in both the numerator and denominator.

In our exercise, the resulting expression is \( \frac{6rb + 4sb^2}{24b^3} \). Given that the denominator \( 24b^3 \) was not the least common denominator, there might be opportunities to simplify further, although sometimes the exercise may not simplify significantly.

Steps to consider for simplification:

• Factor any shared terms in the numerator as much as possible.
• Look for common factors between the numerator and the denominator.

Even if immediate simplification is not obvious, checking the expression for possible reductions is important and ensures precision.

If no further factors are shared, then your fraction is already in its most concise form.