Question
Fill in the blank to make equivalent rational expressions. \( \frac{8}{v+5}=\frac{1}{(v-8)(v+5)} \)
Ask by Newton Wheeler. in the United States
Jan 29,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Fill in the numerator on the right with \(8(v - 8)\) to make both sides equal:
\[
\frac{8}{v+5} = \frac{8(v-8)}{(v-8)(v+5)}
\]
Solution
To make the rational expressions equivalent, you need to ensure that both sides of the equation represent the same value for all permissible values of \( v \).
Given the equation:
\[
\frac{8}{v+5} = \frac{1}{(v-8)(v+5)}
\]
We can manipulate the right side to make it equivalent to the left side by adjusting the numerator. Specifically, the left side has a numerator of 8, whereas the right side has a numerator of 1. To balance the equation, we can multiply the numerator of the right side by 8:
\[
\frac{8}{v+5} = \frac{8(v-8)}{(v-8)(v+5)}
\]
This simplifies to:
\[
\frac{8}{v+5} = \frac{8(v-8)}{(v-8)(v+5)}
\]
Now, both sides of the equation have the same form, making them equivalent.
**Final Answer:**
Fill in the numerator on the right with 8(v – 8) so both sides are equal:
\(\frac{8}{v+5} = \frac{8(v-8)}{(v-8)(v+5)}\)
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To make the expressions equivalent, we need to have the same denominator. The given expressions have the denominator of the left side as \( v + 5 \), and the denominator on the right side as \( (v - 8)(v + 5) \). To achieve this, we can multiply the numerator on the left side by \( (v - 8) \). This gives us: \[\frac{8(v - 8)}{(v + 5)(v - 8)} = \frac{8v - 64}{(v + 5)(v - 8)}\] Now, both sides have the denominator \( (v - 8)(v + 5) \), which makes the expressions equivalent when we write: \[\frac{8(v - 8)}{(v - 8)(v + 5)} = \frac{1}{(v - 8)(v + 5)}\] So, the filled-in blank for the right side to make it equivalent would be \( 8(v-8) \).
