Let's break down the two expressions step by step.

a) $\sqrt{3}(\sqrt{6} + \sqrt{54})$

First, let's simplify the terms inside the parentheses:

1. Simplify $\sqrt{54}$:

   $\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}$

   So the expression becomes:

   $\sqrt{3}(\sqrt{6} + 3\sqrt{6})$

2. Combine like terms inside the parentheses:

   $\sqrt{6} + 3\sqrt{6} = 4\sqrt{6}$

   Now, we have:

   $\sqrt{3} \times 4\sqrt{6}$

3. Multiply the square roots:

   $\sqrt{3} \times \sqrt{6} = \sqrt{3 \times 6} = \sqrt{18}$

   And simplifying $\sqrt{18}$:

   $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

   So the expression becomes:

   $4 \times 3\sqrt{2} = 12\sqrt{2}$

Thus, the simplified expression is:

$12\sqrt{2}$

b) $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})$

This is a product of conjugates, so we can use the difference of squares formula:

$(a - b)(a + b) = a^2 - b^2$

In this case, $a = \sqrt{5}$ and $b = \sqrt{2}$. Applying the formula:

$(\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3$

Thus, the simplified expression is:

$3$

Final Answers:

• a) $12\sqrt{2}$
• b) $3$

Would you like further details or have any questions on this? Here are some related questions for you to explore:

1. What happens when you multiply two square roots that are not conjugates?
2. How do you simplify expressions with cube roots?
3. Can you explain the difference between rationalizing the numerator and denominator?
4. What is the difference between a sum of square roots and a product of square roots?
5. How do you handle higher powers of square roots, like $\sqrt{3^4}$?

Tip: When dealing with square roots, always try to break them into simpler factors (like squares or cubes) to make simplification easier!