Problem 16 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{11 m}{21} \cdot \frac{14}{55 m^{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{15m^2} \).

Step 1: Write the Expression

Start by writing the multiplication of the fractions as one fraction: \[\frac{11m}{21} \times \frac{14}{55m^3} = \frac{11m \times 14}{21 \times 55m^3}\].

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \[\frac{11m \times 14}{21 \times 55m^3} = \frac{154m}{1155m^3}\].

Step 3: Factor and Simplify the Fraction

Identify common factors in the numerator and the denominator. The expression \( m \) appears in both, and the numbers 154 and 1155 are divisible by 11:1. Factor out the common factor \( 11 \) in 154 and 1155. 2. Reduce using common factors:\[\frac{154m}{1155m^3} = \frac{14m}{105m^3}\]3. Simplify \( \frac{m}{m^3} = \frac{1}{m^2} \). After canceling and simplifying the term \( m \), you get:\[\frac{14}{105m^2}\].

Step 4: Further Simplify the Fraction

Simplify \( \frac{14}{105} \) by finding the greatest common divisor (GCD) of 14 and 105, which is 7:\[\frac{14}{105} = \frac{14 \div 7}{105 \div 7} = \frac{2}{15}\].Thus, the fraction becomes:\[\frac{2}{15m^2}\].

Key Concepts

Algebraic Fractions

In algebra, fractions can include not just numbers, but variables as well. These are called algebraic fractions. An algebraic fraction is simply a fraction with variables in the numerator, the denominator, or both. For example, \( \frac{11m}{21} \) is an algebraic fraction. When working with algebraic fractions, it helps to follow similar rules to those used in regular arithmetic fractions.

• Numerators and denominators can include expressions rather than just numbers.
• Variables, such as \( m \), can be treated like numbers for multiplying or simplifying expressions.

Being comfortable with algebraic fractions is essential for tackling more complex algebraic topics. Simplifying these often involves factoring and applying arithmetic operations, all while considering the rules of algebra.

Multiplying Fractions

Multiplying fractions might look tricky at first, but it's actually quite straightforward. Let's break it down. When you multiply fractions, you multiply the numerators together and the denominators together to make a single fraction. For instance, in the expression \( \frac{11m}{21} \times \frac{14}{55m^3} \), you get \( \frac{11m \times 14}{21 \times 55m^3} \). Each fraction's components are combined separately:

• The numerator becomes 154m (from 11m times 14).
• The denominator is 1155m^3 (from 21 times 55m^3).

Make sure to simplify once you've multiplied. This often involves first factoring out any common elements between the numerator and denominator.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form, much like simplifying numeric fractions. A rational expression is essentially a fraction in which both the numerator and the denominator are polynomials. To simplify:

• First, identify and factor out common elements in both the numerator and the denominator.
• Simplify by canceling out similar factors. For instance, in \( \frac{154m}{1155m^3} \), the \( m \) term can be reduced, simplifying to \( \frac{14}{105m^2} \).
• Lastly, reduce the numbers by finding their greatest common divisor (GCD). With GCD being 7 for 14 and 105, it further simplifies to \( \frac{2}{15m^2} \).

This process of simplification ensures the expression is as concise as possible, making further calculations and operations manageable.