Simplify: \( \left|\left(a^{4} b^{5}\right)^{2} \div\left(a^{9} b^{7}\right)^{0}\right| \div a b^{2} \) (3marks)

To simplify the expression \( \left|\left(a^{4} b^{5}\right)^{2} \div\left(a^{9} b^{7}\right)^{0}\right| \div a b^{2} \), we can follow these steps: 1. Simplify the inner expression \( \left(a^{4} b^{5}\right)^{2} \): \[ \left(a^{4} b^{5}\right)^{2} = a^{8} b^{10} \] 2. Simplify the second part \( \left(a^{9} b^{7}\right)^{0} \): \[ \left(a^{9} b^{7}\right)^{0} = 1 \quad (\text{Any expression to the power of 0 is 1}) \] 3. Now, we perform the division: \[ a^{8} b^{10} \div 1 = a^{8} b^{10} \] 4. The absolute value does not change anything since \( a^{8} b^{10} \) is already non-negative. Therefore: \[ \left| a^{8} b^{10} \right| = a^{8} b^{10} \] 5. Next, we divide by \( a b^{2} \): \[ \frac{a^{8} b^{10}}{a b^{2}} = a^{8-1} b^{10-2} = a^{7} b^{8} \] Thus, the simplified expression is: \[ \boxed{a^{7} b^{8}} \]