(4) \( \frac{5^{a+4}-5^{a+3}}{5.5^{a+1}} \)

To simplify the expression \( \frac{5^{a+4}-5^{a+3}}{5.5^{a+1}} \), we can start by factoring the numerator. Notice that both terms in the numerator share a common factor of \( 5^{a+3} \): \[ 5^{a+4} - 5^{a+3} = 5^{a+3}(5 - 1) = 5^{a+3} \cdot 4. \] Now, substituting back into the expression gives us: \[ \frac{5^{a+3} \cdot 4}{5 \cdot 5^{a+1}}. \] Next, we simplify the denominator: \[ 5 \cdot 5^{a+1} = 5^{1 + a + 1} = 5^{a + 2}. \] Now, we have: \[ \frac{4 \cdot 5^{a+3}}{5^{a+2}}. \] Using the properties of exponents, \( \frac{5^{a+3}}{5^{a+2}} = 5^{(a+3)-(a+2)} = 5^{1} = 5 \). So now our expression simplifies to: \[ 4 \cdot 5 = 20. \] Thus, the simplified form of the original expression is \( 20 \).