Let's solve the given problem step-by-step to make the rational expressions equivalent:
Given rational expressions:
[tex]\[
\frac{-7}{2w} = \frac{x}{2w^5}
\][/tex]
Our goal is to find the value of [tex]\(x\)[/tex] that makes these expressions equivalent.
Step 1: Understand that for two rational expressions to be equivalent, their numerators and denominators must be proportional.
The denominator on the right side is [tex]\(2w^5\)[/tex], which involves [tex]\(w^5\)[/tex]. We should manipulate the left side's denominator to also become [tex]\(2w^5\)[/tex] so we can easily compare the numerators.
Step 2: Multiply the numerator and the denominator of the left side expression by [tex]\(w^4\)[/tex]:
[tex]\[
\frac{-7}{2w} \times \frac{w^4}{w^4} = \frac{-7w^4}{2w^5}
\][/tex]
Now, both sides have the same denominator:
[tex]\[
\frac{-7w^4}{2w^5} = \frac{x}{2w^5}
\][/tex]
Step 3: Since the denominators are now equal, we need to make the numerators equal:
[tex]\[
-7w^4 = x
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that makes the original expression equivalent is:
[tex]\[
x = -7w^4
\][/tex]
Hence, the blank should be filled with [tex]\(-7w^4\)[/tex]:
[tex]\[
\frac{-7}{2w} = \frac{-7w^4}{2w^5}
\][/tex]
