To make the fraction \(\frac{2t}{3t-5}\) equivalent with the new denominator \(t(3t-5)(t+3)\), we need to carefully determine what the numerator should be. Let's break it down step-by-step:
1. Identify the Current Numerator and Denominator:
- The current fraction is \(\frac{2t}{3t-5}\).
2. Compare Denominators:
- The current denominator is \(3t-5\).
- The new denominator is \(t(3t-5)(t+3)\).
3. Find the Additional Factor:
- Our goal is to understand what additional factors have been multiplied to the original denominator to get the new denominator.
- The original denominator \(3t-5\) has been multiplied by \(t\) and \((t+3)\).
4. Adjust the Numerator Accordingly:
- Since we have multiplied the denominator by \(t(t+3)\), we should also multiply the numerator by the same factor to maintain the equivalency of the fraction.
5. Compute the New Numerator:
- The original numerator is \(2t\).
- By multiplying this numerator by the additional factors, we get:
[tex]\[
(2t) \times t \times (t+3)
\][/tex]
6. Write Down the New Numerator:
- When you simplify, the new numerator becomes \(2t^2(t+3)\).
Thus, the equivalent fraction with the given denominator is:
[tex]\[
\frac{2t(t+3)}{t(3t-5)(t+3)}
\][/tex]
So, filling in the blank, we get:
[tex]\[
\frac{2t}{3t-5} = \frac{2t(t+3)}{t(3t-5)(t+3)}
\][/tex]
