\frac{3}{m-n}+\frac{m+n}{(m-n)^2}
Answer (2)
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: Expand and combine like terms in the numerator.
Factor the numerator: Factor out common factors from the numerator to obtain the final simplified expression: ( m − n ) 2 2 ( 2 m − n ) .
Explanation
Understanding the Problem 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 rewrite the first fraction with this denominator: m − n 3 = ( m − n ) ( m − n ) 3 ( m − n ) = ( m − n ) 2 3 ( m − n ) .
Adding the Fractions Now we can add the two fractions: ( m − n ) 2 3 ( m − n ) + ( m − n ) 2 m + n = ( m − n ) 2 3 ( m − n ) + ( m + n ) .
Simplifying the Numerator Next, we simplify the numerator by expanding and combining like terms: ( 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 ) . There are no common factors in the numerator and denominator that can be cancelled.
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 complex expressions involving variables like initial velocity, launch angle, and gravitational acceleration. By simplifying these expressions, you can more easily analyze the motion of the projectile and make accurate predictions.
To simplify m − n 3 + ( m − n ) 2 m + n , we find a common denominator of ( m − n ) 2 and rewrite the first fraction. After combining the fractions and simplifying, we arrive at the final expression ( m − n ) 2 2 ( 2 m − n ) .
;
