Simplify.

$\frac{7}{a+8}+\frac{7}{a^2-64}$

Factor the denominator: a 2 − 64 = ( a − 8 ) ( a + 8 ) .
Rewrite the expression with the common denominator: ( a − 8 ) ( a + 8 ) 7 ( a − 8 ) ​ + ( a − 8 ) ( a + 8 ) 7 ​ .
Combine the numerators: ( a − 8 ) ( a + 8 ) 7 ( a − 8 ) + 7 ​ .
Simplify the numerator: ( a − 8 ) ( a + 8 ) 7 a − 56 + 7 ​ = ( a − 8 ) ( a + 8 ) 7 a − 49 ​ .
The simplified expression is ( a − 8 ) ( a + 8 ) 7 a − 49 ​ ​ .

Explanation

Understanding the problem We are asked to simplify the expression a + 8 7 ​ + a 2 − 64 7 ​ and choose the correct answer from the given options.

Factoring the denominator First, we factor the denominator a 2 − 64 as a difference of squares: a 2 − 64 = ( a − 8 ) ( a + 8 ) .

Finding a common denominator Now we rewrite the expression using the factored denominator: a + 8 7 ​ + ( a − 8 ) ( a + 8 ) 7 ​ To add these fractions, we need a common denominator, which is ( a − 8 ) ( a + 8 ) . We rewrite the first fraction with this common denominator: ( a + 8 ) ( a − 8 ) 7 ( a − 8 ) ​ + ( a − 8 ) ( a + 8 ) 7 ​

Combining and simplifying Now we can combine the numerators: ( a − 8 ) ( a + 8 ) 7 ( a − 8 ) + 7 ​ Next, we simplify the numerator: ( a − 8 ) ( a + 8 ) 7 a − 56 + 7 ​ = ( a − 8 ) ( a + 8 ) 7 a − 49 ​

Final Answer Finally, we compare our simplified expression with the given options. The correct answer is: ( a − 8 ) ( a + 8 ) 7 a − 49 ​


Examples
Simplifying rational expressions is a fundamental skill in algebra, useful in various real-world applications. For instance, when designing structures, engineers often deal with complex equations involving rational functions. Simplifying these expressions allows for easier analysis and optimization of the design, ensuring stability and efficiency. Similarly, in economics, simplifying rational expressions can help in modeling supply and demand curves, leading to better predictions and decision-making.

Answered by GinnyAnswer | 2025-07-08