2*2

To solve the expression given in the image: \[ \frac{\sqrt{16a^9 \cdot \sqrt{4b^6}}}{\sqrt{a^3 \cdot b^3}} \] Substitute \(a = 9\) and \(b = 11\) into the expression and then simplify: 1. Simplify the expression: - \(\sqrt{16a^9} = 4a^{4.5}\) - \(\sqrt{4b^6} = 2b^3\) - \(\sqrt{a^3 \cdot b^3} = a^{1.5}b^{1.5}\) 2. Substitute and simplify: \[ \frac{4a^{4.5} \cdot 2b^3}{a^{1.5}b^{1.5}} \] \[ = \frac{8a^{4.5}b^3}{a^{1.5}b^{1.5}} \] \[ = 8a^{(4.5-1.5)}b^{(3-1.5)} \] \[ = 8a^3b^{1.5} \] 3. Substitute \(a = 9\) and \(b = 11\): \[ = 8 \times 9^3 \times 11^{1.5} \] Now, calculate numerically: - \(9^3 = 729\) - \(11^{1.5} \approx 11 \times \sqrt{11} \approx 11 \times 3.317 \approx 36.487\) Calculate: \[ 8 \times 729 \times 36.487 \approx 212,065.416 \] This approximation results from rounding during calculation.