\frac{3}{m-n}+\frac{m+n}{(m-n)^2}
Answer (1)
Find a common denominator: Rewrite the fractions with the common denominator ( m − n ) 2 .
Combine the fractions: Add the numerators over the common denominator.
Simplify the numerator: Combine like terms in the numerator.
Factor the numerator: Factor out common factors from the numerator, resulting in the final simplified expression: ( m − n ) 2 2 ( 2 m − n )
Explanation
Problem Analysis We are asked to simplify the expression m − n 3 + ( m − n ) 2 m + n . To do this, we need to find a common denominator and combine the two fractions.
Finding a Common Denominator The common denominator for the two fractions is ( m − n ) 2 . We need to rewrite the first fraction with this denominator. To do this, we multiply the numerator and denominator of the first fraction by ( m − n ) : m − n 3 × m − n m − n = ( m − n ) 2 3 ( m − n ) = ( m − n ) 2 3 m − 3 n
Adding the Fractions Now we can add the two fractions: ( m − n ) 2 3 m − 3 n + ( m − n ) 2 m + n = ( m − n ) 2 ( 3 m − 3 n ) + ( m + n )
Simplifying the Numerator Combine like terms in the numerator: ( m − n ) 2 3 m − 3 n + m + n = ( m − n ) 2 4 m − 2 n
Factoring the Numerator We can factor out a 2 from the numerator: ( m − n ) 2 2 ( 2 m − n )
Final Answer The simplified expression is ( m − n ) 2 2 ( 2 m − n ) .
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics. It's used in various fields, such as physics, engineering, and computer science. For example, when calculating the trajectory of a projectile, you might need to simplify an expression involving variables like initial velocity, launch angle, and gravitational acceleration. By simplifying the expression, you can make the calculations easier and more efficient.
