Simplify (2/(u+1)+4/(u-8))/(5/(u+1))
You are presented with an algebraic expression that requires simplification. The expression consists of a complex fraction where the numerator is the sum of two fractions, each with different denominators, and the denominator itself is a single fraction. The goal is to simplify this complex fraction into a more manageable form, preferably by combining the fractions in the numerator, simplifying the resulting expression, and then dividing by the fraction in the denominator to obtain a simplified overall result.
Invert the denominator and multiply it with the numerator:
Create a common denominator for
Create a common denominator for
Combine the fractions under a common denominator
Multiply
Multiply
Arrange the factors of
Combine the numerators over the common denominator:
Simplify the numerator.
Extract the common factor of
Extract
Extract
Apply the distributive property:
Multiply
Combine
Combine
Extract the common factor of
Extract
Extract
Extract
Multiply
Simplify the terms.
Eliminate the common factor of
Cancel out the common factor:
Rewrite the simplified expression:
Multiply
Reposition
The problem involves simplifying a complex fraction, which is a fraction where the numerator or the denominator (or both) are also fractions. The process includes finding a common denominator, combining fractions, and simplifying the result.
Key knowledge points include:
Multiplying by the reciprocal to divide fractions.
Finding a common denominator to combine fractions.
Simplifying expressions by factoring out common factors.
Applying the distributive property (a(b + c) = ab + ac).
Canceling common factors in the numerator and denominator.
Understanding that multiplying by a fraction equivalent to 1 does not change the value of an expression.
In this problem, we also see the importance of carefully manipulating algebraic expressions and the utility of factoring to simplify terms. It's crucial to ensure that all steps maintain the equivalence of the original expression to arrive at a simplified form.