Let's go through the questions one by one based on the image you've provided:

1. $3^2 \times 3^{-2} \times 3^3$:

   • Using the laws of exponents, $a^m \times a^n = a^{m+n}$.
   • So, $3^2 \times 3^{-2} = 3^{2 + (-2)} = 3^0 = 1$.
   • Then $1 \times 3^3 = 3^3 = 27$.
   • Answer: 27

2. $8^{\frac{2}{3}}$:

   • This represents the cube root of 8 squared.
   • $8^{\frac{2}{3}} = \left( \sqrt[3]{8} \right)^2 = 2^2 = 4$.
   • Answer: 4

3. $2^2 + 4^2 + 6^2$:

   • Squaring each term:
     • $2^2 = 4$,
     • $4^2 = 16$,
     • $6^2 = 36$.
   • Adding them: $4 + 16 + 36 = 56$.
   • Answer: 56

4. $\sqrt{5} \times 3\sqrt{5} \times 2\sqrt{8}$:

   • Simplify each part:
     • $\sqrt{5} \times 3\sqrt{5} = 3(\sqrt{5})^2 = 3 \times 5 = 15$,
     • $\sqrt{8} = 2\sqrt{2}$, so $2\sqrt{8} = 4\sqrt{2}$.
   • Final expression: $15 \times 4\sqrt{2} = 60\sqrt{2}$.
   • Answer: $60\sqrt{2}$.

5. $\sqrt{10} \times \sqrt{2}$:

   • Multiply under the same square root:
     • $\sqrt{10} \times \sqrt{2} = \sqrt{10 \times 2} = \sqrt{20}$.
   • Simplify $\sqrt{20} = 2\sqrt{5}$.
   • Answer: $2\sqrt{5}$.

6. $8\sqrt{2} + 4\sqrt{2} + 8\sqrt{2}$:

   • Combine like terms (since all are multiples of $\sqrt{2}$):
     • $8\sqrt{2} + 4\sqrt{2} + 8\sqrt{2} = (8 + 4 + 8)\sqrt{2} = 20\sqrt{2}$.
   • Answer: $20\sqrt{2}$.

Would you like further details or explanations on any of these solutions? Here are some additional practice problems:

1. Simplify $3^4 \times 3^{-1} \times 3^2$.
2. Evaluate $16^{\frac{3}{4}}$.
3. What is the sum of squares of $3, 5,$ and $7$?
4. Simplify $2\sqrt{3} \times 5\sqrt{3}$.
5. Multiply $\sqrt{6} \times \sqrt{3}$.

Tip: When multiplying square roots, remember that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$.