\( \frac { 25 \times t ^ { - 4 } } { 5 ^ { - 3 } \times 10 ^ { - 8 } t ^ { - 8 } } \quad ( t \neq 6 ) \)

To simplify the expression \( \frac { 25 \times t ^ { - 4 } } { 5 ^ { - 3 } \times 10 ^ { - 8 } t ^ { - 8 } } \), we first simplify the denominator. Recall that \( 10 = 2 \times 5 \), so \( 10^{-8} = 2^{-8} \times 5^{-8} \). This means: \[ 5^{-3} \times 10^{-8} = 5^{-3} \times 2^{-8} \times 5^{-8} = 2^{-8} \times 5^{-11} \] Now rewriting the whole expression: \[ \frac{25 \times t^{-4}}{2^{-8} \times 5^{-11} \times t^{-8}} \] Simplifying the numerator \( 25 = 5^2 \): \[ = \frac{5^2 \times t^{-4}}{2^{-8} \times 5^{-11} \times t^{-8}} = \frac{5^2 \times t^{-4} \times 2^8 \times 5^{11} \times t^{8}}{1} \] Combining the powers of \( 5 \) and \( t \): \[ = 2^8 \times 5^{2 + 11} \times t^{-4 + 8} = 2^8 \times 5^{13} \times t^{4} \] So, the simplified expression becomes: \[ = 2^8 \times 5^{13} \times t^4 \] Now calculating \( 2^8 = 256 \), the final expression can be presented as: \[ = 256 \times 5^{13} \times t^4 \]