Let's solve each of the inequalities step-by-step based on the image provided.

1. $3\pi$ versus $\sqrt{80}$

• Approximate values:

  • $3\pi \approx 3 \times 3.1416 = 9.4248$
  • $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \approx 4 \times 2.236 = 8.944$

  So, $3\pi > \sqrt{80}$.

2. $\sqrt{10}$ versus $2 \times \sqrt{5}$

• Approximate values:

  • $\sqrt{10} \approx 3.162$
  • $2 \times \sqrt{5} = 2 \times 2.236 \approx 4.472$

  So, $\sqrt{10} < 2 \times \sqrt{5}$.

3. $\sqrt{\frac{1}{10}}$ versus $\sqrt{\frac{1}{8}}$

• Approximate values:

  • $\sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \approx 0.316$
  • $\sqrt{\frac{1}{8}} = \frac{1}{\sqrt{8}} \approx 0.354$

  So, $\sqrt{\frac{1}{10}} < \sqrt{\frac{1}{8}}$.

4. $\pi$ versus $\sqrt{8}$

• Approximate values:

  • $\pi \approx 3.1416$
  • $\sqrt{8} = 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828$

  So, $\pi > \sqrt{8}$.

Summary:

• $3\pi > \sqrt{80}$
• $\sqrt{10} < 2 \times \sqrt{5}$
• $\sqrt{\frac{1}{10}} < \sqrt{\frac{1}{8}}$
• $\pi > \sqrt{8}$

Would you like a more detailed explanation for any of these comparisons?

Here are 5 related questions you might find helpful:

1. How do you approximate square roots without a calculator?
2. What is the relationship between powers of 10 and square roots?
3. How do you compare irrational numbers like $\pi$ and $e$?
4. What are common approximations used for $\pi$ in geometry problems?
5. How can we simplify square roots of fractions?

Tip: Always check for simplifications when comparing square roots, as this can make mental math easier!