Simplification and Value Calculation

a) $\frac{2}{\sqrt{5}}$

To simplify $\frac{2}{\sqrt{5}}$, we rationalize the denominator:

$\frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$

Using $\sqrt{5} = 2.236$: $\frac{2\sqrt{5}}{5} = \frac{2 \times 2.236}{5} = \frac{4.472}{5} \approx 0.894$

Thus, $\frac{2}{\sqrt{5}} \approx 0.894$.

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b) $\sqrt{8} + 4\sqrt{32} - 6\sqrt{2}$

1. Simplify each term:

   • $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
   • $4\sqrt{32} = 4 \times \sqrt{16 \times 2} = 4 \times 4\sqrt{2} = 16\sqrt{2}$
   • $6\sqrt{2}$ remains as is.

2. Combine like terms: $\sqrt{8} + 4\sqrt{32} - 6\sqrt{2} = 2\sqrt{2} + 16\sqrt{2} - 6\sqrt{2}$

$= (2 + 16 - 6)\sqrt{2} = 12\sqrt{2}$

3. Substitute $\sqrt{2} = 1.414$: $12\sqrt{2} = 12 \times 1.414 = 16.968$

Thus, $\sqrt{8} + 4\sqrt{32} - 6\sqrt{2} \approx 16.968$.

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Final Results:

• a) $\frac{2}{\sqrt{5}} \approx 0.894$
• b) $\sqrt{8} + 4\sqrt{32} - 6\sqrt{2} \approx 16.968$

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Would you like me to explain any step in more detail or verify further?

Relative Questions:

1. How can you rationalize a denominator?
2. Why is combining like terms important in simplifying expressions?
3. Can you generalize how to handle square roots in simplifications?
4. How do approximations affect calculations in real-world problems?
5. What are some common strategies to simplify radical expressions?

Tip: Always check for opportunities to factorize terms involving square roots to simplify calculations!