Problem 34 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{a b^{2}}{c} \cdot \frac{c^{2}}{a^{2} b}$$

Step-by-Step Solution

Answer

\( \frac{cb}{a} \)

Step 1: Setup the Multiplication

Start by writing the multiplication of the two fractions as a single fraction: \( \frac{a b^{2}}{c} \cdot \frac{c^{2}}{a^{2} b} = \frac{a b^{2} \cdot c^{2}}{c \cdot a^{2} b} \).

Step 2: Simplify the Numerator and Denominator

Multiply the numerators and denominators: \(\text{Numerator: } a \cdot b^{2} \cdot c^{2} = a c^{2} b^{2} \)\(\text{Denominator: } c \cdot a^{2} \cdot b = a^{2} b c \)

Step 3: Simplify Common Factors

Cancel out the common factors in the numerator and denominator. Cancel \(a\):\[ \frac{a c^{2} b^{2}}{a^{2} b c} = \frac{c^{2} b^{2}}{a b c} \]Cancel \(b\):\[ \frac{c^{2} b}{a c} \]Cancel \(c\):\[ \frac{c b}{a} \]

Step 4: Write the Simplified Answer

Write the fraction after all cancellations: \( \frac{c b}{a} \). This is the fraction in its lowest terms.

Key Concepts

Multiplying Fractions

When you multiply fractions, you are essentially combining two rational expressions into one. Imagine you have two fractions, say \( \frac{a}{b} \) and \( \frac{c}{d} \). To multiply them, you simply find the product of the numerators and the product of the denominators:

• Numerator: \( a \times c \)
• Denominator: \( b \times d \)

This results in the fraction \( \frac{a \cdot c}{b \cdot d} \).
It's important to keep fractions in their simplest form whenever possible. This means reducing them by finding and canceling any common factors in the numerator and denominator after multiplication.
Multiplying fractions provides a perfect opportunity to learn about simplifying expressions. It requires careful attention to detail to ensure each part of the fraction is reduced as fully as possible before seeking the final simplest form.

Rational Expressions

Rational expressions are similar to fractions but involve polynomials in the numerator and/or the denominator. They play a crucial role in algebra and higher mathematics.
A rational expression looks something like \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. The key is ensuring the denominator is never zero, as dividing by zero is undefined.

• When dealing with rational expressions, always look to factorize the polynomials.
• Factorization allows you to see common factors more easily, facilitating simplification.

When you multiply rational expressions, follow the same rules as you do when multiplying regular fractions. This means multiplying the numerators to form a new numerator and the denominators to form a new denominator.
The clarity provided by understanding rational expressions helps with both simplifying them and performing operations like multiplication, division, addition, or subtraction.

Common Factors

Common factors are key to simplifying both fractions and rational expressions. They are numbers or expressions that divide evenly into both the numerator and the denominator.
To identify common factors, look for numbers, variables, or even entire expressions that repeat in both parts of the fraction. For instance, if you have a fraction like \( \frac{2x^2}{4x} \), you can see that \( 2x \) is a common factor. By dividing both the numerator and the denominator by \( 2x \), you simplify it to \( \frac{x}{2} \).

• Always ensure you've identified all common factors before canceling. This means factoring completely first.
• Once identified, cancel the common factors to reduce the expression, making sure not to cancel terms that only appear once.

Simplifying expressions by canceling common factors is a foundational skill in algebra. It paves the way for more complex operations and provides deeper understanding in reducing expressions and solving equations.