Here’s a breakdown of each problem in the image. I'll start with the first few problems:

1. Последовательно упростите

$\sqrt{7 + 3\sqrt{5} - 2\sqrt{11 - 4\sqrt{3}}}$

To solve this, we need to simplify the inner expressions step-by-step, focusing on nested square roots. We can rewrite or simplify expressions inside the square roots where possible.

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2. Последовательно упростите

$\sqrt{4 - \sqrt{15 + 2\sqrt{14 + 3\sqrt{6}}}}$

This also involves simplifying the expressions inside the nested square roots in sequence. We need to evaluate the inner terms first, moving outward.

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3. Вычислите

$\frac{15 \cdot (\sqrt{5} + 2\sqrt{6} + 3)^2}{(\sqrt{3} + 1) \cdot (\sqrt{3} - 1)}$

Here, simplify the numerator by expanding $(\sqrt{5} + 2\sqrt{6} + 3)^2$ and then simplify the denominator using the difference of squares formula.

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4. Вычислите

$\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}}$

Rationalize each term separately by multiplying the numerator and denominator by the conjugate of the denominator, then add the results.

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5. Вычислите

$\sqrt{21 - 4\sqrt{30} \cdot (\sqrt{15} + 2\sqrt{2})}$

Simplify by evaluating $\sqrt{30} \cdot (\sqrt{15} + 2\sqrt{2})$ first and then simplifying the expression inside the square root.

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Would you like a full solution to each problem or a specific question answered?

5 Related Questions:

1. How would you approach simplifying nested square roots in general?
2. What strategies are helpful in rationalizing denominators involving square roots?
3. Why is the difference of squares useful in simplifying these types of expressions?
4. How can you verify if a square root simplification is correct?
5. What are some common techniques for simplifying expressions with multiple radicals?

Tip:

When simplifying expressions with nested radicals, try to work from the innermost radical outward to avoid mistakes in the simplification process.