We can solve the expression $(\sqrt{5} + \sqrt{3})(3\sqrt{2} - 5\sqrt{6})$ by using the distributive property (also known as the FOIL method for binomials). Let's break it down step by step:

The expression is:

$(\sqrt{5} + \sqrt{3})(3\sqrt{2} - 5\sqrt{6})$

Step 1: Distribute each term in the first parentheses to each term in the second parentheses.

• First, distribute $\sqrt{5}$ to both terms in the second binomial: $\sqrt{5} \cdot 3\sqrt{2} = 3\sqrt{10}$ $\sqrt{5} \cdot (-5\sqrt{6}) = -5\sqrt{30}$

• Next, distribute $\sqrt{3}$ to both terms in the second binomial: $\sqrt{3} \cdot 3\sqrt{2} = 3\sqrt{6}$ $\sqrt{3} \cdot (-5\sqrt{6}) = -5\sqrt{18}$

Step 2: Simplify the square roots.

• $\sqrt{18}$ simplifies as follows: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ So, $-5\sqrt{18}$ becomes: $-5\sqrt{18} = -5 \times 3\sqrt{2} = -15\sqrt{2}$

Step 3: Combine all the terms.

Now, let's combine all the results from the distribution: $3\sqrt{10} - 5\sqrt{30} + 3\sqrt{6} - 15\sqrt{2}$

So, the final expanded expression is:

$3\sqrt{10} - 5\sqrt{30} + 3\sqrt{6} - 15\sqrt{2}$

This is the simplified result of the product.

Would you like more details or have any questions about this solution?

Here are 5 related questions you can explore:

1. How can we simplify square roots like $\sqrt{30}$ or $\sqrt{18}$ further?
2. Can you explain how the distributive property works with binomials in more detail?
3. What is the significance of simplifying square roots in expressions?
4. Can you solve other algebraic expressions involving square roots?
5. How do we handle binomials with more than two terms?

Tip: When multiplying square roots, remember to combine like terms and simplify where possible. If a term inside the square root is a product, you can break it down into simpler square roots.