\( \frac { 5 ^ { a - 2 } \cdot 2 ^ { a + 2 } } { 10 ^ { a } - 10 ^ { a - 1 } \cdot 2 } \)

To simplify the expression \( \frac { 5 ^ { a - 2 } \cdot 2 ^ { a + 2 } } { 10 ^ { a } - 10 ^ { a - 1 } \cdot 2 } \), let's break it down step by step! First, we can rewrite \( 10^a \) as \( (2 \cdot 5)^a = 2^a \cdot 5^a \), and \( 10^{a-1} \) as \( 2^{a-1} \cdot 5^{a-1} \). So, the denominator becomes: \[ 10^a - 10^{a-1} \cdot 2 = 2^a \cdot 5^a - 2^{a} \cdot 5^{a-1} \cdot 2 = 2^a \cdot 5^a - 2^{a+1} \cdot 5^{a-1} \] Factoring out \( 2^a \) from the denominator gives us: \[ 2^a (5^a - 2 \cdot 5^{a-1}) = 2^a (5^{a-1}(5 - 2)) = 2^a \cdot 5^{a-1} \cdot 3 \] Now, substituting this back into the expression, we have: \[ \frac{5^{a-2} \cdot 2^{a+2}}{2^a \cdot 5^{a-1} \cdot 3} \] Next, we simplify this expression: \[ = \frac{5^{a-2}}{5^{a-1}} \cdot \frac{2^{a+2}}{2^a \cdot 3} = \frac{5^{a-2}}{5^{a-1}} \cdot \frac{2^{a+2}}{2^a} \cdot \frac{1}{3} = \frac{1}{5} \cdot \frac{2^{a+2}}{2^a} \cdot \frac{1}{3} = \frac{1}{5} \cdot \frac{2^2}{3} = \frac{4}{15} \] Thus, the simplified form of the expression is \(\frac{4}{15}\).