Let's evaluate the expressions step by step to find the correct one.

A:

$\sqrt{\frac{8}{2}} + \sqrt{\frac{16}{4}} + \sqrt{\frac{24}{6}}$

• $\sqrt{\frac{8}{2}} = \sqrt{4} = 2$
• $\sqrt{\frac{16}{4}} = \sqrt{4} = 2$
• $\sqrt{\frac{24}{6}} = \sqrt{4} = 2$

Thus, the expression in option A evaluates to: $2 + 2 + 2 = 6$

B:

$(\sqrt{4^2}) + \sqrt{16} + 6^3$

• $\sqrt{4^2} = \sqrt{16} = 4$
• $\sqrt{16} = 4$
• $6^3 = 216$

Thus, the expression in option B evaluates to: $4 + 4 + 216 = 224$

C:

$(4^2 + 5^3) + \sqrt{100} + \frac{181}{6}$

• $4^2 = 16$
• $5^3 = 125$
• $\sqrt{100} = 10$
• $\frac{181}{6} \approx 30.17$

Thus, the expression in option C evaluates to: $(16 + 125) + 10 + 30.17 = 151 + 30.17 = 181.17$

D:

$\sqrt{\frac{8}{3}} + \sqrt{\frac{16}{3}} + \sqrt{\frac{24}{3}}$

• $\sqrt{\frac{8}{3}} \approx 1.63$
• $\sqrt{\frac{16}{3}} \approx 2.31$
• $\sqrt{\frac{24}{3}} \approx 2.83$

Thus, the expression in option D evaluates to: $1.63 + 2.31 + 2.83 = 6.77$

Summary of Results:

• A = 6
• B = 224
• C = 181.17
• D = 6.77

The correct value for 6 is found in option A.

Do you want any further details or clarifications on these steps? Here are some related questions:

1. How do you simplify square roots with fractions inside them?
2. What is the general rule for simplifying cube roots?
3. Why does $\sqrt{4^2} = 4$ instead of 16?
4. How do you calculate fractional expressions with square roots?
5. What are the steps for evaluating complex expressions with powers and roots?

Tip: Always simplify the fractions first before applying square roots to make the process easier.