The expression given in the image is:

$\frac{15(\sqrt{10} + \sqrt{5})}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}}$

We need to simplify this expression.

Step 1: Simplify the square roots

Let's simplify the square roots inside the denominator:

• $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$
• $\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$
• $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$

So, the denominator becomes:

$\sqrt{10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5}$

Step 2: Combine like terms in the denominator

Now, let's combine the like terms:

• The $\sqrt{10}$ terms: $\sqrt{10} + 2\sqrt{10} = 3\sqrt{10}$
• The $\sqrt{5}$ terms: $2\sqrt{5} - \sqrt{5} - 4\sqrt{5} = -3\sqrt{5}$

So, the denominator simplifies to:

$3\sqrt{10} - 3\sqrt{5}$

Step 3: Simplify the entire expression

Now, substitute the simplified denominator back into the original expression:

$\frac{15(\sqrt{10} + \sqrt{5})}{3(\sqrt{10} - \sqrt{5})}$

We can factor out the 3 from the denominator:

$\frac{15(\sqrt{10} + \sqrt{5})}{3} \times \frac{1}{(\sqrt{10} - \sqrt{5})}$

This simplifies to:

$5(\sqrt{10} + \sqrt{5}) \times \frac{1}{(\sqrt{10} - \sqrt{5})}$

Step 4: Rationalize the denominator

Now, multiply the numerator and denominator by the conjugate of the denominator $\sqrt{10} + \sqrt{5}$:

$\frac{5(\sqrt{10} + \sqrt{5}) \times (\sqrt{10} + \sqrt{5})}{(\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5})}$

The denominator simplifies using the difference of squares:

$(\sqrt{10})^2 - (\sqrt{5})^2 = 10 - 5 = 5$

So, the expression simplifies to:

$\frac{5 \times (\sqrt{10} + \sqrt{5})^2}{5}$

Canceling the 5:

$(\sqrt{10} + \sqrt{5})^2$

Now expand the square:

$(\sqrt{10})^2 + 2(\sqrt{10})(\sqrt{5}) + (\sqrt{5})^2 = 10 + 2\sqrt{50} + 5$

Simplify further:

$15 + 10\sqrt{5}$

Thus, the simplified expression is:

$\boxed{5(3 + 2\sqrt{5})}$ which matches option (c) in the given choices.

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Do you want more details or have any questions?

Here are five related questions:

1. How do you simplify expressions involving square roots in the numerator and denominator?
2. What is the purpose of rationalizing the denominator?
3. Can you explain how to use the difference of squares to simplify expressions?
4. How does factoring out common terms help in simplifying fractions?
5. What other techniques can be used to simplify complex algebraic expressions?

Tip: When rationalizing the denominator, always look for the conjugate pair to eliminate the square roots in the denominator.