-(1%2F5))%2F((1%2Fx%5E2)-(1%2F25)).jpeg "((1/x)-(1/5))/((1/x^2)-(1/25))")
Simplifying the Expression: ((1/x)-(1/5))/((1/x^2)-(1/25))
This expression involves fractions within fractions, also known as complex fractions. To simplify it, we'll employ a few algebraic techniques.
1. Combining Fractions in the Numerator and Denominator
First, let's combine the fractions in the numerator and denominator separately.
Numerator:
- Find a common denominator for (1/x) and (1/5), which is 5x.
- ((1/x)-(1/5)) = (5/5x) - (x/5x) = (5-x)/5x
Denominator:
- Find a common denominator for (1/x^2) and (1/25), which is 25x^2.
- ((1/x^2)-(1/25)) = (25/25x^2) - (x^2/25x^2) = (25-x^2)/25x^2
2. Dividing by a Fraction
Now, our expression looks like this: ((5-x)/5x) / ((25-x^2)/25x^2)
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of ((25-x^2)/25x^2) is (25x^2/(25-x^2)).
Therefore:
((5-x)/5x) * (25x^2/(25-x^2))
3. Simplifying the Expression
- Notice that (25-x^2) can be factored as (5+x)(5-x).
- Cancel out the common factors (5-x) and 5x.
The final simplified expression is:
(5x)/(5+x)
Important Note: This simplification is valid as long as x ≠ 0 and x ≠ -5. These values would make the original expression undefined.