Simplify. Express your answer using positive exponents. \( \frac{7 s t^{0} u}{7 s^{3} t u \cdot s^{7} t u} \)

To simplify the expression \( \frac{7 s t^{0} u}{7 s^{3} t u \cdot s^{7} t u} \), we start by recognizing that \( t^{0} = 1 \). Therefore, the expression simplifies to: \( \frac{7 s u}{7 s^{3} t u \cdot s^{7} t u} \). Now, we'll simplify the denominator: \( 7 s^{3} t u \cdot s^{7} t u = 7 s^{3} s^{7} t^2 u^2 = 7 s^{10} t^2 u^2 \). Now the overall expression is: \( \frac{7 s u}{7 s^{10} t^2 u^2} \). Next, we can cancel the \( 7 \) in the numerator and denominator: \( \frac{s u}{s^{10} t^2 u^2} \). Next, we can cancel \( u \) from the numerator and \( u^2 \) from the denominator, yielding: \( \frac{s}{s^{10} t^2 u} \). Now we simplify \( \frac{s}{s^{10}} = s^{1-10} = s^{-9} \). Thus, we have: \( \frac{s^{-9}}{t^2 u} \). Expressing it using positive exponents, we write: \( \frac{1}{s^9 t^2 u} \). So the final simplified expression is: \(\frac{1}{s^9 t^2 u}\).