Let's solve these problems step by step:

Problem 18:

$6\sqrt{2} - 3\sqrt{2} + 8\sqrt{2}$

• Combine like terms:
  $(6 - 3 + 8)\sqrt{2} = 11\sqrt{2}$

Answer: $11\sqrt{2}$

Problem 20:

$5\sqrt{3} - 2\sqrt{12}$

• Simplify $\sqrt{12}$:
  $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$
• Substitute back:
  $5\sqrt{3} - 2(2\sqrt{3}) = 5\sqrt{3} - 4\sqrt{3}$
• Combine like terms:
  $(5 - 4)\sqrt{3} = \sqrt{3}$

Answer: $\sqrt{3}$

Problem 22:

$(6 + \sqrt{2})(3 - \sqrt{2})$
Use the distributive property:
$6 \cdot 3 + 6(-\sqrt{2}) + \sqrt{2} \cdot 3 + \sqrt{2}(-\sqrt{2})$
Simplify step by step:

• $6 \cdot 3 = 18$
• $6(-\sqrt{2}) = -6\sqrt{2}$
• $\sqrt{2} \cdot 3 = 3\sqrt{2}$
• $\sqrt{2}(-\sqrt{2}) = -2$

Combine terms:
$18 - 6\sqrt{2} + 3\sqrt{2} - 2$
$(18 - 2) + (-6\sqrt{2} + 3\sqrt{2}) = 16 - 3\sqrt{2}$

Answer: $16 - 3\sqrt{2}$

Problem 24:

$(2 + \sqrt{3})^2$
Expand using the formula $(a + b)^2 = a^2 + 2ab + b^2$:
$2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2$
Simplify step by step:

• $2^2 = 4$
• $2(2)(\sqrt{3}) = 4\sqrt{3}$
• $(\sqrt{3})^2 = 3$

Combine terms:
$4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$

Answer: $7 + 4\sqrt{3}$

If you’d like further explanation, let me know! Here are 5 related questions to practice:

1. How would you simplify $7\sqrt{5} + 2\sqrt{20}$?
2. Expand and simplify $(3 - \sqrt{2})^2$.
3. Simplify $4\sqrt{27} - 3\sqrt{12}$.
4. What is $\sqrt{8} + \sqrt{18} - \sqrt{50}$?
5. Simplify $(1 + \sqrt{3})(2 - \sqrt{3})$.

Tip: Always look for perfect squares within radicals to simplify terms effectively.