$\frac{(m+n)^2}{m^2-n^2}+\frac{m^2+m n}{n^2-m n}$
Answer (1)
m 2 − n 2 ( m + n ) 2 + n 2 − mn m 2 + mn simplifies as follows:
Factor the denominators: m 2 − n 2 = ( m − n ) ( m + n ) and n 2 − mn = n ( n − m ) .
Simplify the expression: ( m − n ) ( m + n ) ( m + n ) 2 + n ( n − m ) m ( m + n ) = m − n m + n − n ( m − n ) m ( m + n ) .
Combine the fractions: n ( m − n ) n ( m + n ) − m ( m + n ) = n ( m − n ) ( m + n ) ( n − m ) .
Further simplification yields: − n m + n = − n m − 1 . Thus, the final answer is − n m + n .
Explanation
Analyze the problem We are asked to simplify the expression m 2 − n 2 ( m + n ) 2 + n 2 − mn m 2 + mn .
To simplify this expression, we will first factor the denominators and then find a common denominator to combine the two fractions.
Factor the denominators First, we factor the denominators:
m 2 − n 2 = ( m − n ) ( m + n ) and n 2 − mn = n ( n − m ) .
So the expression becomes
( m − n ) ( m + n ) ( m + n ) 2 + n ( n − m ) m 2 + mn
Simplify the terms Next, we simplify the first term:
( m − n ) ( m + n ) ( m + n ) 2 = m − n m + n
And we factor the numerator of the second term:
n ( n − m ) m 2 + mn = n ( n − m ) m ( m + n )
Rewrite with common denominator Now we rewrite the second term to have a common denominator. Notice that n − m = − ( m − n ) , so
n ( n − m ) m ( m + n ) = − n ( m − n ) m ( m + n ) = − n ( m − n ) m ( m + n )
Thus, the expression becomes
m − n m + n − n ( m − n ) m ( m + n )
Combine the fractions Now we find a common denominator for both terms, which is n ( m − n ) :
n ( m − n ) n ( m + n ) − n ( m − n ) m ( m + n ) = n ( m − n ) n ( m + n ) − m ( m + n )
Factor the numerator We factor out ( m + n ) from the numerator:
n ( m − n ) ( m + n ) ( n − m )
Simplify the expression Finally, we simplify the expression. Notice that n − m = − ( m − n ) , so
n ( m − n ) ( m + n ) ( n − m ) = n ( m − n ) ( m + n ) ( − ( m − n )) = − n m + n
We can rewrite this as
− n m + n = − n m − n n = − n m − 1
So the simplified expression is
− n m − 1
State the final answer Therefore, the simplified expression is
− n m − 1
We can also write this as
− n m + n
Examples
Simplifying rational expressions is a fundamental skill in algebra, with applications in various fields. For instance, in physics, you might encounter complex fractions when dealing with electrical circuits or fluid dynamics. Simplifying these expressions allows for easier analysis and calculation of relevant quantities, such as current, voltage, or flow rates. In economics, rational expressions can appear in cost-benefit analyses or when modeling supply and demand curves. By simplifying these expressions, economists can gain insights into market behavior and make informed decisions.
