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.