Problem 45 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{5 b}{24 a^{2}}-\frac{11 a}{32 b} $$

Step-by-Step Solution

Answer

\(\frac{40b^2 - 66a^3}{192a^2b}\)

Step 1: Find Common Denominator

To add or subtract rational expressions, we first need to find a common denominator. The denominators here are \(24a^2\) and \(32b\). The least common multiple (LCM) of these denominators is \(192a^2b\).

Step 2: Adjust Each Fraction

Rewrite each fraction with the common denominator \(192a^2b\). Multiply \(\frac{5b}{24a^2}\) by \(\frac{8b}{8b}\) to get \(\frac{40b^2}{192a^2b}\). Multiply \(\frac{11a}{32b}\) by \(\frac{6a^2}{6a^2}\) to get \(\frac{66a^3}{192a^2b}\).

Step 3: Subtract the Fractions

Now that the denominators are the same, subtract the numerators: \(\frac{40b^2 - 66a^3}{192a^2b}\).

Step 4: Simplify the Expression

Factor out any common terms in the numerator if possible and check for any common factors in the entire fraction. However, \(40b^2 - 66a^3\) does not have common factors with the denominator \(192a^2b\), so the expression is already in its simplest form.

Key Concepts

Common Denominator

When dealing with rational expressions, think of the common denominator as a shared language that allows fractions to "communicate" or be combined. In this exercise, to subtract fractions such as \(\frac{5b}{24a^2}\) and \(\frac{11a}{32b}\), we needed a common denominator. This helps us represent both fractions without changing their value, but in a format that makes combining them possible.

A common denominator must be a multiple of each individual denominator. This ensures that each original fraction can be adjusted or rewritten using the common denominator.

Common Denominator: Finding It

• List the multiples of each denominator.
• The smallest multiple that appears in both lists becomes your common denominator.

In this case, the common denominator was found to be \(192a^2b\). This single denominator acts as a bridge, allowing you to seamlessly subtract the two rational expressions.

Least Common Multiple

The least common multiple (LCM) is an essential concept in creating a common denominator. It refers to the smallest number that both denominators (in this case) can be divided into without a remainder. Remember, identifying the LCM needs a structured process:

• Break down each denominator into its prime factors.
• Multiply each factor the greatest number of times it occurs in any of the factors' prime factorizations.

For example, to find the LCM of \(24a^2\) and \(32b\), we can break them down as:

• \(24a^2 = 2^3 \times 3 \times a^2\)
• \(32b = 2^5 \times b\)

The LCM is then \(2^5 \times 3 \times a^2 \times b = 192a^2b\). This thorough approach guarantees that both denominators can transform into an equivalent expression with the LCM as their new denominator.

Simplifying Fractions

Once you've combined fractions over a common denominator, the next step is simplification. Simplifying fractions involves reducing them to their simplest form, making them easier to understand and work with.Steps to Simplify:

• Factor out any common factors in the numerator and the denominator if present.
• Cancel out these common factors.

In our problem, the expression was \(\frac{40b^2 - 66a^3}{192a^2b}\), and since no common factors exist between the numerator \(40b^2 - 66a^3\) and the denominator \(192a^2b\), further simplification wasn't necessary. Remember to always double-check for common factors to ensure the fraction is completely simplified. This keeps your work neat and precise, helping you in reaching more accurate results efficiently.