To solve the first three questions, we will use the properties of exponents.
For the expression \( y^8 \cdot y^{-2} \), we will apply the rule that states when multiplying like bases, you add the exponents.
For the expression \( \left(x^4\right)^{-3} \), we will use the power of a power rule, which involves multiplying the exponents.
For the expression \( \left(65^4 t^{-7}\right)^2 \), we will apply the power of a product rule, which involves distributing the exponent to each factor inside the parentheses.
Step 1: Simplifying \( y^8 \cdot y^{-2} \)
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can simplify:
\[
y^8 \cdot y^{-2} = y^{8 + (-2)} = y^6
\]
Step 2: Simplifying \( \left(x^4\right)^{-3} \)
Applying the power of a power rule, which states \( (a^m)^n = a^{m \cdot n} \), we have:
\[
\left(x^4\right)^{-3} = x^{4 \cdot (-3)} = x^{-12}
\]
Using the power of a product rule, which states \( (ab)^n = a^n b^n \), we can distribute the exponent:
\[
\left(65^4 t^{-7}\right)^2 = 65^{4 \cdot 2} \cdot t^{-7 \cdot 2} = 65^8 \cdot t^{-14}
\]
Final Answer
The results for each expression are:
\( y^6 \)
\( x^{-12} \)
\( 65^8 \cdot t^{-14} \)
Thus, the final answers are:
\[
\boxed{y^6}, \quad \boxed{x^{-12}}, \quad \boxed{65^8 \cdot t^{-14}}
\]