To simplify $(t^3)^5$ using the given base $t$, apply the power of a power rule, which states:

$(a^m)^n = a^{m \cdot n}$

In this case: $(t^3)^5 = t^{3 \cdot 5} = t^{15}$

So, the simplified form is: $t^{15}$

Would you like a further breakdown of this rule or have any additional questions?

Here are some related questions to expand on this concept:

1. How would you simplify $(x^4)^6$?
2. What is the general formula for $(a^m)^n$?
3. How would you express $(y^2)^8$ in exponential form?
4. What happens if the exponents are fractions, such as $(z^{1/2})^4$?
5. Can you explain why the power of a power rule works?

Tip: Always multiply the exponents when raising a power to another power, keeping the base unchanged.