Find the product \(\frac{8}{6 n-4} \cdot\left(9 n^2-4\right)\).
An expression can be written in rational form by writing it as a fraction with a denominator of what?
Answer (2)
12 n + 8
Explanation
Understanding the Problem We are asked to find the product of a rational expression and a polynomial expression. The expression is 6 n − 4 8 ⋅ ( 9 n 2 − 4 ) . We can factor the denominator 6 n − 4 and the term 9 n 2 − 4 to simplify the expression.
Factoring the Denominator First, let's factor the denominator 6 n − 4 . We can factor out a 2: 6 n − 4 = 2 ( 3 n − 2 ) .
Factoring the Numerator Next, let's factor the term 9 n 2 − 4 . This is a difference of squares, so we can factor it as: 9 n 2 − 4 = ( 3 n − 2 ) ( 3 n + 2 ) .
Rewriting the Expression Now, let's rewrite the original expression with the factored forms: 6 n − 4 8 ⋅ ( 9 n 2 − 4 ) = 2 ( 3 n − 2 ) 8 ⋅ ( 3 n − 2 ) ( 3 n + 2 ) .
Cancelling Common Factors We can cancel the common factor ( 3 n − 2 ) from the numerator and denominator: 2 ( 3 n − 2 ) 8 ⋅ ( 3 n − 2 ) ( 3 n + 2 ) = 2 8 ⋅ ( 3 n + 2 ) .
Simplifying Now, simplify the expression by dividing 8 by 2: 2 8 ⋅ ( 3 n + 2 ) = 4 ( 3 n + 2 ) .
Distributing Finally, distribute the 4 to get the simplified expression: 4 ( 3 n + 2 ) = 12 n + 8 .
Final Answer The product 6 n − 4 8 ⋅ ( 9 n 2 − 4 ) simplifies to 12 n + 8 . An expression can be written in rational form by writing it as a fraction with a denominator of 1.
Examples
Understanding how to simplify rational expressions is useful in many areas, such as physics and engineering, where complex equations often need to be simplified to make calculations easier. For example, when calculating the trajectory of a projectile, you might encounter rational expressions that need to be simplified to find the optimal launch angle. Similarly, in electrical engineering, simplifying rational expressions can help in analyzing circuits and determining the values of components. This skill is also useful in computer graphics, where rational expressions are used to model curves and surfaces.
The product 6 n − 4 8 ⋅ ( 9 n 2 − 4 ) simplifies to 12 n + 8 . An expression can be written in rational form with a denominator of 1. Thus, the final form is 12 n + 8 .
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