Fully simplify the expression below. \[ \left(5 t^{6} u^{8}\right)^{2} \div 30 t^{3} u^{2} \] Give your answer as a fraction without any negative indices.

To simplify the expression \(\left(5 t^{6} u^{8}\right)^{2} \div 30 t^{3} u^{2}\), we start by simplifying the numerator. 1. **Expand the numerator:** \[ \left(5 t^{6} u^{8}\right)^{2} = 5^{2} (t^{6})^{2} (u^{8})^{2} = 25 t^{12} u^{16} \] 2. **Set up the division:** Now, we rewrite the division as multiplication by the reciprocal: \[ 25 t^{12} u^{16} \div 30 t^{3} u^{2} = 25 t^{12} u^{16} \cdot \frac{1}{30 t^{3} u^{2}} = \frac{25 t^{12} u^{16}}{30 t^{3} u^{2}} \] 3. **Simplify the coefficients:** \[ \frac{25}{30} = \frac{5}{6} \] 4. **Simplify the variable parts:** For \(t\): \[ \frac{t^{12}}{t^{3}} = t^{12-3} = t^{9} \] For \(u\): \[ \frac{u^{16}}{u^{2}} = u^{16-2} = u^{14} \] 5. **Combine the simplified parts:** Putting it all together: \[ \frac{5}{6} t^{9} u^{14} \] Thus, the fully simplified expression is: \[ \frac{5 t^{9} u^{14}}{6} \]