Problem 21 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{9 a^{2} c}{12 b c^{2}} \div \frac{21 a b}{14 c^{3}}\)

Step-by-Step Solution

Answer

\( \frac{a c^2}{2 b} \)

Step 1: Understand the Division of Fractions

Division of fractions involves multiplying by the reciprocal. Given the problem \( \frac{9 a^{2} c}{12 b c^{2}} \div \frac{21 a b}{14 c^{3}} \), rewrite it as a multiplication problem: \( \frac{9 a^{2} c}{12 b c^{2}} \times \frac{14 c^{3}}{21 a b} \).

Step 2: Multiply the Numerators and Denominators

Multiply the numerators: \( 9a^2c \times 14c^3 = 126a^2c^4 \). Multiply the denominators: \( 12bc^2 \times 21ab = 252a b^2c^2 \).

Step 3: Simplify the Expression

Cancel out the common factors in the numerator and denominator. Start with the numerical coefficients: \( 126 \div 252 = \frac{1}{2} \). Next, cancel the variable terms: \( \frac{a^2}{a} = a \), \( \frac{c^4}{c^2} = c^2 \). Consequently, \( \frac{126a^2c^4}{252ab^2c^2} = \frac{a c^2}{2 b} \).

Step 4: Express in Simplest Form

Ensure there are no further common factors to cancel. The simplest form of the expression is \( \frac{a c^2}{2 b} \).

Key Concepts

Division of Fractions

When faced with the division of fractions, it can initially seem complex, but the key is to remember that dividing by a fraction is the same as multiplying by its reciprocal.

• To find the reciprocal of a fraction, simply swap the numerator and the denominator.
• This means that instead of dividing, you'll multiply by this flipped version.

In the example exercise involving rational expressions, the division process—\(\frac{9 a^{2} c}{12 b c^{2}} \div \frac{21 a b}{14 c^{3}}\)—is changed into a multiplication problem: \(\frac{9 a^{2} c}{12 b c^{2}} \times \frac{14 c^{3}}{21 a b}\). This transformation forms the foundation of solving division problems involving fractions. It converts the operation into something more straightforward: multiplication.

Simplifying Expressions

Simplifying expressions is like organizing math neatly. It involves reducing an expression to its most basic form, ensuring that it's easier to read and understand.

• The process includes canceling out terms that appear in both the numerator and denominator.
• This step is crucial for transforming complicated fractions into simpler ones.

For the given problem, once we rewrote it as a multiplication problem, we ended up with new fractions \(\frac{126a^2c^4}{252ab^2c^2}\). To simplify, start by handling the numbers: dividing 126 and 252 by their greatest common divisor. Here, the fraction reduces from \(\frac{126}{252}\) to \(\frac{1}{2}\). Then, move on to the variables:- Cancel common variable terms. For instance, \(a^2\) over \(a\) simplifies to \(a\), and \(c^4\) over \(c^2\) becomes \(c^2\). - This process gives the final simplified expression \(\frac{ac^2}{2b}\).Simplifying ensures you have the cleanest version of the expression.

Multiplying Fractions

Multiplying fractions is an essential and straightforward process in rational expressions.

• Multiply the numerators to get the new numerator.
• Multiply the denominators to find the new denominator.

Continuing from where we reciprocated the divisor, after rewriting the division problem, we have \(\frac{9 a^{2} c}{12 b c^{2}} \times \frac{14 c^{3}}{21 a b}\).The multiplication is done step by step: - First, multiply the top parts of the fractions (numerators): \(9a^2c \times 14c^3 = 126a^2c^4\).- Next, multiply the bottom parts (denominators): \(12bc^2 \times 21ab = 252ab^2c^2\).These calculations lay the ground for the simplification step. Remember, multiplication sets up the numbers and variables for reducing, but it always comes first before simplifying. The overall process ensures that fractions are combined correctly into a single coherent expression.