#### Answer

It can be seen that Shannon incorrectly took the square roots.
Therefore, the correct answer is $2a^{2}\sqrt[3] {2b^{10}}$.

#### Work Step by Step

Step 1: We will solve the expression to find the correct answer and to determine Shannon's mistake.
Step 2: $\sqrt[3] {16a^{6}b^{10}}$
Step 3: $\sqrt[3] {16}\times\sqrt[3] {a^{6}}\times\sqrt[3] {b^{10}}$
Step 4: $\sqrt[3] {2\times8}\times(a)^{6\times\frac{1}{3}}\times (b)^{10\times\frac{1}{3}}$
Step 5: $\sqrt[3] {2}\times\sqrt[3] 8\times(a)^{2}\times (b)^{\frac{10}{3}}$
Step 6: $\sqrt[3] {2}\times2\times a^{2}\times \sqrt[3] {b^{10}}$
Step 7: Therefore, the correct answer is $2a^{2}\sqrt[3] {2b^{10}}$
Step 8: It can be seen that Shannon incorrectly took the square roots.