Explain why this statement is true: when simplifying rational expressions, we must cancel factors only, and not terms. Give an example using just numbers.
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The statement is true because when simplifying rational expressions, we can cancel common factors in the numerator and the denominator, but we cannot cancel common terms.
To understand this, consider the following example:
Let's simplify the rational expression (4x + 8)/(2x + 4).
If we try to cancel the term (4x) in both the numerator and denominator, we would end up with (4x + 8)/(2x + 4) = 1. However, this would not be correct.
Instead, we need to factor out the common factor in the numerator and the denominator, which is 4.
(4x + 8)/(2x + 4) = 4(x + 2)/4(2x + 4)
Now, we can cancel the common factor of 4:
4(x + 2)/4(2x + 4) = (x + 2)/(2x + 4)
In this example, we canceled the factor 4 correctly, but we did not cancel the terms (x + 2) in the numerator and (2x + 4) in the denominator because they are not common factors. Only common factors can be canceled when simplifying rational expressions.
When simplifying rational expressions, it is important to cancel factors only and not terms. This is because factors can be canceled out, whereas terms cannot.
To understand this concept, let's consider an example with numbers.
Consider the rational expression (6x)/(3y). If we want to simplify this expression, we need to look for common factors in the numerator and denominator that can be canceled out.
In this case, we have the factor "3" in both the numerator (6x) and the denominator (3y). By canceling out this common factor, we can simplify the expression as follows:
(6x)/(3y) = (2 * 3 * x)/(3 * y)
= (2 * (3x))/(3y)
As you can see, the common factor of 3 in both the numerator and the denominator allows us to cancel it out. This simplification is valid because dividing both the numerator and denominator by the same non-zero number does not change the value of the expression.
However, if we had terms instead of factors, canceling them would not be valid. Let's consider another example: (2x + 3)/(x + 5).
In this case, we have the terms "2x" and "3" in the numerator, and the terms "x" and "5" in the denominator. If we try to cancel out the "x" term in the numerator and denominator, we would obtain:
(2x + 3)/(x + 5) = (2 * x + 3)/(x + 5)
= (2 + 3)/(1 + 5)
= 5/6
This is incorrect because canceling out terms in the numerator and denominator is not a valid operation in algebraic simplification. We can only cancel out common factors.
In summary, when simplifying rational expressions, we can cancel out common factors in the numerator and denominator, but we should not cancel out terms. Cancelling factors is a valid operation because it does not change the value of the expression, whereas canceling terms can lead to incorrect results.