Question
Fill in the blank to make equivalent rational expressio \( \frac{-9}{2 x}=\frac{\square}{2 x^{5}} \)
Ask by Salinas Ramirez. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To make the rational expressions equivalent, the blank should be filled with \(-9x^4\). So, the equation becomes:
\[
\frac{-9}{2x} = \frac{-9x^4}{2x^5}
\]
Solution
We start with the given equality:
(-9)/(2x) = (□)/(2x⁵)
To find what should replace the square (□), we'll equate the two expressions.
1. Multiply both sides of the equation by 2x⁵ to eliminate the denominators:
(-9)/(2x) × 2x⁵ = (□)/(2x⁵) × 2x⁵
This simplifies to:
-9x⁵/x = □
2. Simplify the expression on the left:
-9x⁵/x = -9x^(5-1) = -9x⁴
Thus, the numerator that makes the rational expressions equivalent is:
□ = -9x⁴
So the equivalent expression is:
(-9)/(2x) = (-9x⁴)/(2x⁵)
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Simplify this solution Bonus Knowledge
To make the rational expressions equivalent, you need to maintain the same value on both sides of the equation. If you multiply the numerator and denominator of the left-hand fraction by \(x^4\), you'll have: \[ \frac{-9 \cdot x^4}{2 x \cdot x^4} = \frac{-9 x^4}{2 x^5} \] So, the blank should be filled in with \(-9 x^4\). Thus, the equivalent expression is: \[ \frac{-9}{2 x} = \frac{-9 x^4}{2 x^5} \]
