Problem 20 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{3}{10 a}-\frac{13}{15 a^{3}} $$

Step-by-Step Solution

Answer

\(\frac{9a^2 - 26}{30a^3}\) is the simplified expression.

Step 1: Identify the Least Common Denominator (LCD)

The denominators we have are \(10a\) and \(15a^3\). To find the LCD, we need to determine the least common multiple (LCM) of the coefficients 10 and 15, and also manage the variable part. The LCM of 10 and 15 is 30. The highest power of \(a\) present is \(a^3\). Therefore, the LCD is \(30a^3\).

Step 2: Rewrite Each Fraction with the LCD

Now we rewrite each fraction with the common denominator \(30a^3\).For \(\frac{3}{10a}\), multiply the numerator and the denominator by \(3a^2\) to get:\[\frac{3(3a^2)}{10a(3a^2)} = \frac{9a^2}{30a^3}\]For \(\frac{13}{15a^3}\), multiply the numerator and the denominator by \(2\) to get:\[\frac{13 \times 2}{15a^3 \times 2} = \frac{26}{30a^3}\]

Step 3: Combine the Fractions

Since both fractions now have the common denominator, we can combine them:\[\frac{9a^2}{30a^3} - \frac{26}{30a^3} = \frac{9a^2 - 26}{30a^3}\]

Step 4: Simplify if Possible

In the expression \(\frac{9a^2 - 26}{30a^3}\), check if any further simplification of the numerator is possible. Here, \(9a^2 - 26\) and \(30a^3\) have no common factor other than 1. Therefore, the expression is already in its simplest form and can't be simplified further.

Key Concepts

Least Common Denominator

When dealing with algebraic fractions, finding the least common denominator (LCD) is an essential step. It helps in aligning fractions to the same base for easier manipulation. The LCD is essentially the least common multiple (LCM) of the denominators involved.
To find the LCD of the fractions \(\frac{3}{10a}\) and \(\frac{13}{15a^3}\), we focus on two key components:

• The coefficients: Here, the coefficients are 10 and 15. The LCM of these numbers is 30, as 30 is the smallest number that both 10 and 15 can divide without leaving a remainder.
• The variable part: We look for the highest power of the variable present in the denominators. In this case, since one of the denominators already includes \(a^3\), we take \(a^3\) as the highest power.

The LCD, therefore, becomes \(30a^3\). This common denominator brings uniformity to the fractions, setting the stage for further calculations.

Simplifying Expressions

Simplifying expressions involves making an expression as concise as possible. This often involves reducing fractions to their lowest terms or combining like terms.
Once the common denominator of \(30a^3\) is found for both fractions, we need to rewrite these fractions with the new common denominator:

• For \(\frac{3}{10a}\), multiply both the numerator and denominator by \(3a^2\) to adjust it to \(\frac{9a^2}{30a^3}\).
• For \(\frac{13}{15a^3}\), multiply both the numerator and denominator by 2, providing \(\frac{26}{30a^3}\).

By doing this, we ensure that both fractions align under the same denominator, allowing us to handle them collectively in the next steps. This step is crucial as it paves the way to easily combine and further simplify the expression if possible.

Combining Fractions

Combining fractions becomes straightforward once you've established a common denominator. It involves performing the indicated operations on the numerators while keeping the denominator constant.
In this exercise, we place our focus on:

• Subtracting: With common denominators, subtract the numerators. Thus, \(\frac{9a^2}{30a^3} - \frac{26}{30a^3}\) becomes \(\frac{9a^2 - 26}{30a^3}\).

Once you have the combined fraction, the final step is to check if further simplification is possible. In this case, \(9a^2 - 26\) and \(30a^3\) don’t have common factors other than 1. This confirms that the expression is already in its simplest form.
Combining fractions is a powerful technique to consolidate complex operations into a simpler, single expression, thus making further computations manageable.