Problem 22 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{3 a b^{3}}{4 c} \div \frac{21 a c}{12 b c^{3}}\)

Step-by-Step Solution

Answer

\( \frac{3b^4c^2}{7} \)

Step 1: Understand the Division of Fractions

The operation involves dividing two fractions. Dividing by a fraction is the same as multiplying by its reciprocal. Thus, \( \frac{3ab^3}{4c} \div \frac{21ac}{12bc^3} \) becomes \( \frac{3ab^3}{4c} \times \frac{12bc^3}{21ac} \).

Step 2: Multiply the Fractions

Now, multiply the numerators and the denominators of the fractions: \( \frac{3ab^3 \times 12bc^3}{4c \times 21ac} \), which simplifies to \( \frac{36ab^4c^3}{84ac^2} \).

Step 3: Factor and Simplify the Fraction

Factor both the numerator and the denominator to simplify the fraction. The numerator \(36ab^4c^3\) can be factored as \(36ab \cdot b^3 \cdot c^3\). The denominator \(84ac^2\) can be factored as \(84a \cdot c^2\).

Step 4: Cancel Common Factors

Cancel out the common factors from both the numerator and the denominator: \( \frac{36ab \cdot b^3 \cdot c^3}{84a \cdot c^2} \). The \(a\) in the numerator and denominator cancel, as do one \(c\), leaving us with \( \frac{36b^4c^2}{84} \).

Step 5: Simplify Further

Now, simplify \( \frac{36b^4c^2}{84} \) by dividing both numerator and denominator by their greatest common divisor, which is 12. This results in \( \frac{3b^4c^2}{7} \).

Step 6: Final Simplification

Ensure all terms are simplified to their lowest terms. The fraction \( \frac{3b^4c^2}{7} \) is already in its simplest form.

Key Concepts

Fraction Division

Fraction division is an important concept in algebra, especially when handling rational expressions. The essential rule for dividing fractions is quite straightforward. Instead of dividing directly, you multiply by the reciprocal.
For example, given two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the division can be transformed into a multiplication: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \).
Applying this knowledge to rational expressions involves the same principles. You simply flip the second fraction and switch the operation from division to multiplication. This can greatly simplify complex expressions by reducing the task to something more familiar—multiplication.

Factoring

Factoring plays a crucial role when simplifying rational expressions. It involves breaking down expressions into products of their simplest factors. This can be numbers or variables, and ensures that you're working with the simplest terms at every step.
Consider the expression \( 36ab^4c^3 \). You can factor it stepwise as follows:

• First look at the coefficients: 36 is factored into \(2^2 \times 3^2 \).
• Next, address the variables: break \(ab^4c^3\) down to \(a \cdot b^4 \cdot c^3\).

Through factoring, large and seemingly complex expressions become easier to manage and reduce correctly.

Simplification

Simplification of rational expressions means making them as straightforward as possible, without changing the value. This process often involves factoring and reducing terms.
When you multiply fractions, often you'll find common terms that you can cancel to simplify.
For example, starting with the expression \(\frac{36b^4c^2}{84}\), you simplify by dividing both the numerator and denominator by their greatest common divisor (GCD), which in this case is 12:

• Divide 36 by 12 to get 3.
• Divide 84 by 12 to get 7.

The simplified form becomes \(\frac{3b^4c^2}{7}\), which has been reduced to its simplest terms.

Common Factors

Common factors are shared terms that appear in both the numerator and the denominator of a fraction. Identifying and canceling common factors is key to simplifying an expression.
If we examine \( \frac{36ab^4c^3}{84ac^2} \), we find the common factors:

• The term \(a\) is in both the numerator and the denominator.
• Similarly, \(c^2\) is common but in different powers (\(c^3\) and \(c^2\)).

Cancel these common terms across the fraction, which can often significantly simplify the expression. This practice ensures that you present the rational expression in its lowest terms, making computations easier and more accurate.