\frac{4 m-9 n}{16 m^2}-\frac{9 n^2+1}{4 m-3 n}
Answer (1)
Find a common denominator: 16 m 2 ( 4 m − 3 n ) .
Rewrite the fractions with the common denominator.
Combine the fractions into a single fraction.
Simplify the numerator and the expression: 16 m 2 ( 4 m − 3 n ) 3 n ( − 48 m 2 n − 16 m + 9 n ) .
Explanation
Problem Analysis We are given the expression 16 m 2 4 m − 9 n − 4 m − 3 n 9 n 2 + 1 and we want to simplify it.
Finding Common Denominator First, we find a common denominator for the two fractions. The common denominator is 16 m 2 ( 4 m − 3 n ) . We rewrite each fraction with this common denominator: 16 m 2 ( 4 m − 3 n ) ( 4 m − 9 n ) ( 4 m − 3 n ) − 16 m 2 ( 4 m − 3 n ) ( 9 n 2 + 1 ) ( 16 m 2 )
Combining Fractions Now, we combine the fractions into a single fraction: 16 m 2 ( 4 m − 3 n ) ( 4 m − 9 n ) ( 4 m − 3 n ) − ( 9 n 2 + 1 ) ( 16 m 2 )
Expanding the Numerator Next, we simplify the numerator by expanding and combining like terms: \begin{align*} (4 m-9 n)(4 m-3 n)-(9 n^2+1)(16 m^2) &= 16m^2 - 12mn - 36mn + 27n^2 - (144m^2n^2 + 16m^2) \ &= 16m^2 - 48mn + 27n^2 - 144m^2n^2 - 16m^2 \ &= -48mn + 27n^2 - 144m^2n^2 \end{align*}
Rewriting the Expression So the expression becomes: 16 m 2 ( 4 m − 3 n ) − 48 mn + 27 n 2 − 144 m 2 n 2
Factoring the Numerator We can factor out 3 n from the numerator: 16 m 2 ( 4 m − 3 n ) 3 n ( − 16 m + 9 n − 48 m 2 n ) So the simplified expression is: 16 m 2 ( 4 m − 3 n ) 3 n ( − 48 m 2 n − 16 m + 9 n )
Final Answer Therefore, the simplified expression is 16 m 2 ( 4 m − 3 n ) 3 n ( − 48 m 2 n − 16 m + 9 n )
Examples
Rational expressions are useful in many fields, such as physics, engineering, and economics. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves. Simplifying rational expressions makes it easier to analyze and understand these models. For instance, consider a scenario where you are trying to optimize the design of a bridge. Simplifying rational expressions that describe the forces acting on the bridge can help you determine the optimal dimensions and materials to use.
