(1/x^2-49)/(1/x+7)?
To simplify the expression (1/x^2-49)/(1/x+7), we can follow these steps:
Step 1: Factor the denominators.
The denominator of the first fraction, x^2 - 49, can be factored as a difference of squares: (x - 7)(x + 7).
The denominator of the second fraction, 1/x + 7, does not need to be factored further.
Step 2: Rewrite the expression.
Substituting the factored denominators, the expression becomes:
[(1)/(x - 7)(x + 7)] / (1/x + 7)
Step 3: Simplify the complex fraction.
To simplify a complex fraction like this, we can use the rule that division is equivalent to multiplying by the reciprocal.
So, we can rewrite the complex fraction as follows:
[(1)/(x - 7)(x + 7)] * (x/1)
Notice that we took the reciprocal of the second fraction 1/(1/x + 7), which becomes x.
Step 4: Simplify further.
Now, we can cancel out the common factors between the numerator and denominator:
[(1) * (x)] / [(x - 7)(x + 7)]
This simplifies to:
x / [(x - 7)(x + 7)]
Therefore, the simplified expression is x / [(x - 7)(x + 7)].