Problem 11 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{18 a^{2}}{45 a b}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2a}{5b}\).

Step 1: Factor Numerator and Denominator

To simplify the rational expression, first factor both the numerator and the denominator. The numerator is \(18a^2\), which can be factored as \(2 \times 3^2 \times a^2\). The denominator, \(45ab\), can be factored as \(3^2 \times 5 \times a \times b\).

Step 2: Cancel Common Factors

Next, identify and cancel any common factors in the numerator and the denominator. Both the numerator \(18a^2\) and the denominator \(45ab\) share the common factors \(3^2\) and \(a\). After canceling these, we are left with \(\frac{2a}{5b}\).

Step 3: Write the Simplified Expression

After canceling the common factors, write down the simplified expression. The simplified form of the rational expression is \(\frac{2a}{5b}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a strategy to rewrite expressions as a product of their components. In our example, we have the polynomial expressions in the numerator and denominator of a rational expression, \(\frac{18a^2}{45ab}\). Students often find it helpful to break everything down into prime numbers and variables. Here's how it works:

• First, take the number 18 in the numerator: It's made up of 2 and two 3s, or \(2 \times 3^2\).
• Next, include the variables: \(a^2\) remains as is.

Once the numerator is factored as \(2 \times 3^2 \times a^2\), we move to the denominator:

• The number 45 is made up of two 3s and a 5, or \(3^2 \times 5\).
• Then bring in the variables: Add \(a \times b\) to it.

Thus, the denominator becomes \(3^2 \times 5 \times a \times b\). With these steps, we have prepared the polynomial for the next stage, canceling common factors.

Canceling Common Factors

Canceling common factors is a vital step in simplifying rational expressions. Once you've factored each part of your expression fully, like in \(\frac{18a^2}{45ab}\), you look for identical factors in both the numerator and the denominator. Here's how:

• Both top and bottom have \(3^2\) - these cancel each other because dividing any number by itself equals 1.
• Both have at least one \(a\); cancel it as well by retaining only the necessary power.

So, once the \(3^2\) and \(a\) factors are canceled, the remaining parts are \(2a\) in the numerator and \(5b\) in the denominator. This leaves you with a much simpler expression, \(\frac{2a}{5b}\), making it easier to work with and understand.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions where the numerator and/or the denominator are polynomials. They are a foundational concept in algebra and are essential when working with more complicated equations.

• In our expression \(\frac{18a^2}{45ab}\), both the numerator and denominator fit this description, involving polynomials with constants and variables.
• The simplification process involves factoring, canceling, and rewriting, just as with numerical fractions, but with added complexity due to variables.

Simplifying algebraic fractions is about efficiency. By reducing the fraction, as we did to get \(\frac{2a}{5b}\), you minimize the chance of error in subsequent calculations and make it easier to identify potential mathematical relationships and solutions. Such skills are invaluable in both pure and applied mathematics disciplines.