To express the given difference in its simplest form, we proceed as follows:
[tex]\[
\frac{3 t^2}{5}-\frac{4 t^2}{15}
\][/tex]
First, we need a common denominator to subtract these fractions. The denominators are 5 and 15, and the least common multiple (LCM) of these numbers is 15. Therefore, we need to express [tex]\(\frac{3 t^2}{5}\)[/tex] with a denominator of 15:
[tex]\[
\frac{3 t^2}{5} = \frac{3 t^2 \cdot 3}{5 \cdot 3} = \frac{9 t^2}{15}
\][/tex]
Now, rewrite the original subtraction problem using the common denominator:
[tex]\[
\frac{9 t^2}{15} - \frac{4 t^2}{15}
\][/tex]
Since the denominators are now the same, we can subtract the numerators directly:
[tex]\[
\frac{9 t^2 - 4 t^2}{15} = \frac{5 t^2}{15}
\][/tex]
Next, we simplify the fraction [tex]\(\frac{5 t^2}{15}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[
\frac{5 t^2 \div 5}{15 \div 5} = \frac{t^2}{3}
\][/tex]
Thus, the simplest form of the given expression [tex]\(\frac{3 t^2}{5} - \frac{4 t^2}{15}\)[/tex] is:
[tex]\[
\boxed{\frac{t^2}{3}}
\][/tex]
From the given answer choices, the correct choice is:
C. [tex]\(\frac{t^2}{3}\)[/tex]
