Describe the mathematic process of canceling like factors when working with rational expressions. Demonstrate this with an example
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When working with rational expressions, canceling like factors is a process that involves simplifying the expression by removing common factors from both the numerator and the denominator. This simplification helps to make the expression easier to work with and understand.
Here is a step-by-step explanation of how to cancel like factors:
1. Start by factoring both the numerator and the denominator of the rational expression. Factorization involves breaking down each expression into its prime factors.
2. Identify common factors between the numerator and the denominator. Common factors are those that appear in both the numerator and the denominator.
3. Cancel out these common factors by dividing them out. This entails dividing each common factor in both the numerator and the denominator, effectively removing them from the expression.
4. After canceling out the common factors, you should be left with a simplified rational expression.
Now, let's demonstrate this process with an example:
Consider the rational expression (6x^2y)/(12xy^2).
Step 1: Factor the numerator and the denominator:
The numerator (6x^2y) can be factored as 2 * 3 * x * x * y.
The denominator (12xy^2) can be factored as 2 * 2 * 3 * x * y * y.
Step 2: Identify the common factors:
The common factors between the numerator and the denominator are 2, 3, x, and y.
Step 3: Cancel out the common factors:
By dividing out the common factors simultaneously, we cancel them out:
(2 * 3 * x * x * y)/(2 * 2 * 3 * x * y * y) = (x * x)/(2 * y).
Step 4: Simplify the expression:
The expression (x * x)/(2 * y) is the simplified form after canceling like factors.
To summarize, canceling like factors in rational expressions involves factoring the numerator and denominator, identifying common factors, canceling them out, and simplifying the expression. This process helps to simplify and clarify the rational expression.