To simplify \( \frac{16^5}{4} \), start by rewriting \( 16 \) as a power of \( 4 \):
\[ 16 = 4^2 \]
Now substitute \( 16 \) in the expression:
\[ 16^5 = (4^2)^5 \]
Using the power of a power property, \( (a^m)^n = a^{m \cdot n} \):
\[ (4^2)^5 = 4^{2 \cdot 5} = 4^{10} \]
Now, we can rewrite the original expression:
\[ \frac{16^5}{4} = \frac{4^{10}}{4} \]
Using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[ \frac{4^{10}}{4^1} = 4^{10-1} = 4^9 \]
Now, we can leave the expression as \( 4^9 \) or calculate its value. Calculating \( 4^9 \):
\[ 4^9 = (2^2)^9 = 2^{2 \cdot 9} = 2^{18} \]
Calculating \( 2^{18} \):
\[ 2^{18} = 262144 \]
Thus, the simplified expression can be represented as:
\[ \frac{16^5}{4} = 4^9 = 262144 \]
In conclusion, the simplified form is:
\[ 4^9 \quad \text{or} \quad 262144 \]