Simplify the following: [tex]$\frac{4 n 9^9}{16 n^2}-\frac{2 n^2+1}{4 n-3 n}$[/tex]
Answer (1)
The expression simplifies as follows:
Simplify the first term: 16 n 2 4 n 9 9 = 4 n 9 9
Simplify the second term: 4 n − 3 n 2 n 2 + 1 = n 2 n 2 + 1
Combine the terms with a common denominator: 4 n 9 9 − n 2 n 2 + 1 = 4 n 9 9 − 4 ( 2 n 2 + 1 )
Simplify the numerator: 4 n 387420485 − 8 n 2 .
4 n 387420485 − 8 n 2
Explanation
Initial Analysis We are asked to simplify the expression 16 n 2 4 n 9 9 − 4 n − 3 n 2 n 2 + 1 We will simplify each term separately and then combine them.
Simplifying the First Term First, let's simplify the first term: 16 n 2 4 n 9 9 = 16 4 ⋅ n 2 n ⋅ 9 9 = 4 1 ⋅ n 1 ⋅ 9 9 = 4 n 9 9 Since 9 9 = 387420489 , the first term simplifies to 4 n 387420489
Simplifying the Second Term Next, let's simplify the second term. The denominator is 4 n − 3 n = n , so the second term is n 2 n 2 + 1 Now we have the expression 4 n 387420489 − n 2 n 2 + 1
Combining the Terms To combine the two terms, we need a common denominator, which is 4 n . We rewrite the second term with the common denominator: n 2 n 2 + 1 = 4 n 4 ( 2 n 2 + 1 ) = 4 n 8 n 2 + 4 So the expression becomes 4 n 387420489 − 4 n 8 n 2 + 4 Now we can combine the fractions: 4 n 387420489 − ( 8 n 2 + 4 ) = 4 n 387420489 − 8 n 2 − 4 = 4 n 387420485 − 8 n 2
Final Result Therefore, the simplified expression is 4 n 387420485 − 8 n 2
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics, with applications in various fields such as physics, engineering, and computer science. For instance, when designing a bridge, engineers use simplified equations to calculate the forces acting on the structure. Similarly, in computer graphics, simplifying complex equations can optimize rendering performance, allowing for smoother and more efficient animations. By mastering simplification techniques, students can develop a strong foundation for tackling more advanced mathematical problems and real-world applications.
