Sure! Let's solve for the unknown number in the rational expression provided:
Given:
[tex]\[
\frac{5}{2 v^6} = \frac{\text{?}}{10 v^7}
\][/tex]
We need to find the unknown number such that both sides of the equation are equal.
Step-by-Step:
1. Let's denote the unknown number as [tex]\(x\)[/tex]. Therefore, the equation becomes:
[tex]\[
\frac{5}{2 v^6} = \frac{x}{10 v^7}
\][/tex]
2. To solve for [tex]\(x\)[/tex], we can cross multiply to get rid of the denominators:
[tex]\[
5 \cdot 10 v^7 = x \cdot 2 v^6
\][/tex]
3. Simplify the multiplication on each side:
[tex]\[
50 v^7 = 2 x v^6
\][/tex]
4. To isolate [tex]\(x\)[/tex], divide both sides of the equation by [tex]\(2 v^6\)[/tex]:
[tex]\[
x = \frac{50 v^7}{2 v^6}
\][/tex]
5. Now, simplify the fraction:
[tex]\[
x = \frac{50}{2} \cdot \frac{v^7}{v^6}
\][/tex]
6. This further simplifies to:
[tex]\[
x = 25 \cdot v^1
\][/tex]
Since [tex]\(v^7 / v^6 = v^{7-6} = v^1\)[/tex]
7. Therefore:
[tex]\[
x = 25v
\][/tex]
The unknown number is [tex]\(25v\)[/tex]. So, the equivalent rational expression is:
[tex]\[
\frac{5}{2 v^6} = \frac{25v}{10 v^7}
\][/tex]
