To find the numerical equivalent of \( 5^{10} \cdot 5^{-6} \), we can use the property of exponents which states that when multiplying expressions with the same base, we add the exponents:
\[ a^m \cdot a^n = a^{m+n} \]
In this case, our base \( a \) is \( 5 \), \( m = 10 \), and \( n = -6 \). Thus, we can add the exponents:
\[ 5^{10} \cdot 5^{-6} = 5^{10 + (-6)} = 5^{10 - 6} = 5^4 \]
Next, we calculate \( 5^4 \):
\[ 5^4 = 5 \cdot 5 \cdot 5 \cdot 5 \]
Calculating this step-by-step:
- \( 5 \cdot 5 = 25 \)
- \( 25 \cdot 5 = 125 \)
- \( 125 \cdot 5 = 625 \)
Thus,
\[ 5^4 = 625 \]
Therefore, the numerical equivalent of \( 5^{10} \cdot 5^{-6} \) is
\[ \boxed{625} \]