Problem 7 - Algebra and Trigonometry

Question

\(7-20=\) Simplify the rational expression. $$ \frac{12 x}{6 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2}{x}\).

Step 1: Identify Common Factors

Both the numerator and the denominator have a common factor. Identify the common factor between 12 and 6, which is 6, and notice that the denominator also has a factor of \(x\).

Step 2: Factor Out the Common Factor

Factor 6 out of both the numerator and the denominator. The expression becomes: \[\frac{12x}{6x^2} = \frac{6 \cdot 2x}{6 \cdot x \cdot x}\]

Step 3: Cancel the Common Factors

Cancel the common factors from the numerator and the denominator. The 6's and an \(x\) from each side cancel each other out, simplifying the expression to: \[\frac{2}{x}\]

Step 4: Verify the Simplification

Check that the simplified expression \(\frac{2}{x}\) is in its simplest form, ensuring all common factors have been canceled. There are no further common factors to cancel.

Key Concepts

Common Factor Identification

Before diving into simplifying a rational expression, it's essential to spot any common factors present in both the numerator and the denominator. Common factors are numbers or variables that appear in both components of the expression. In this process, identifying these factors is the crucial first step because it sets the foundation for simplifying the expression later on.

Let's consider an example: for the fraction \( \frac{12x}{6x^2} \), both the number 12 in the numerator and the 6 in the denominator have a common factor of 6. Similarly, the variables "x" (in the numerator) and "x^2" (in the denominator) have an "x" in common. Finding these common factors is like uncovering the hidden building blocks that can be used to rewrite the expression more simply.

To accurately identify these factors, you can use these basic principles:

• Look for the greatest common divisor (GCD) of the coefficients. In our example, the GCD of 12 and 6 is 6.
• Identify the common variables. The smallest power of "x" present in both numerator and denominator is "x".

Finding these common factors makes the next steps a breeze.

Factoring Expressions

After identifying the common factors, the next crucial step in simplification is factoring the expression. Factoring involves rewriting the original expression, breaking it into its elemental parts. By doing so, it becomes effortless to spot the elements that can be cancelled out in the later steps.

For instance, in the expression \( \frac{12x}{6x^2} \), after identifying the common factor to be 6 and "x", we can write the expression as: \[ \frac{6 \cdot 2x}{6 \cdot x \cdot x} \]. This new form clearly shows each factor separately, making it easier to visualize what needs to be simplified.

Here are some simple tips that help with this process:

• Remember the expressions of prime factorization: Break down numbers into prime numbers.
• For variables, list them using exponents. For example, \(x^2 = x \cdot x\).

With the expression factored out, the path to simplification becomes wide open.

Cancelling Common Factors

Now that the common factors have been recognized and the expression has been factored, the simplification journey takes us to the final step: canceling common factors. Canceling is an elegant move that helps reduce the expression to its simplest form.

In our running example with the expression \( \frac{6 \cdot 2x}{6 \cdot x \cdot x} \), you can see both the numerator and denominator share common factors of 6 and one "x". By striking off these common elements—a process known as "canceling"—we distill the expression to its simplest terms: \( \frac{2}{x} \).

Consider these key points when canceling:

• Cancel only factors, not terms. This means anything that multiplies across the whole fraction can be canceled.
• If a common factor appears in both the numerator and denominator, it can be eliminated.

Once the canceling is done, you're left with the simplest possible form of the rational expression, here: \( \frac{2}{x} \). This form is efficient and straightforward, perfectly demonstrating the power of simplification.