To find the product of \( 2^5 \cdot 2^5 \) using the Product Rule of Exponents, we can use the rule that states:
\[ a^m \cdot a^n = a^{m+n} \]
In this case, we have:
- \( a = 2 \)
- \( m = 5 \)
- \( n = 5 \)
Applying the Product Rule:
\[ 2^5 \cdot 2^5 = 2^{5+5} = 2^{10} \]
Now, we can calculate \( 2^{10} \):
\[ 2^{10} = 1024 \]
Thus, the product of \( 2^5 \cdot 2^5 \) is
\[ \boxed{1024} \]