) 13sqrt(7) + 10sqrt(28) - 20sqrt(1, 75) 5) (x ^ 2 + 1 - x * sqrt(3))(x ^ 2 + 1 + x * sqrt(3)) B) (5 - sqrt(3)) ^ 2 + (2 - 3sqrt(3)) ^ 2 - (2sqrt(3) - 1) ^ 2 г 1sqrt(28 -…

Let's solve each problem step by step.

1) Simplify \( 13\sqrt{7} + 10\sqrt{28} - 20\sqrt{1.75} \)

First, simplify the square roots:

• \( \sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7} \)
• \( \sqrt{1.75} = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2} \)

Now substitute these back into the expression: \[ 13\sqrt{7} + 10(2\sqrt{7}) - 20\left(\frac{\sqrt{7}}{2}\right) \] This simplifies to: \[ 13\sqrt{7} + 20\sqrt{7} - 10\sqrt{7} \]

Combine the terms: \[ (13 + 20 - 10)\sqrt{7} = 23\sqrt{7} \]

Answer: \( 23\sqrt{7} \)

5) Expand \((x^2 + 1 - x\sqrt{3})(x^2 + 1 + x\sqrt{3})\)

This expression is a difference of squares: \[ (a - b)(a + b) = a^2 - b^2 \text{ where } a = x^2 + 1 \text{ and } b = x\sqrt{3} \] Calculating \( a^2 \) and \( b^2 \): \[ a^2 = (x^2 + 1)^2 = x^4 + 2x^2 + 1 \] \[ b^2 = (x\sqrt{3})^2 = 3x^2 \]

Putting it into the difference of squares formula: \[ (x^2 + 1)^2 - (x\sqrt{3})^2 = (x^4 + 2x^2 + 1) - 3x^2 = x^4 - x^2 + 1 \]

Answer: \( x^4 - x^2 + 1 \)

B) Simplify \((5 - \sqrt{3})^2 + (2 - 3\sqrt{3})^2 - (2\sqrt{3} - 1)^2\)

Calculate each square:

1. \((5 - \sqrt{3})^2 = 25 - 10\sqrt{3} + 3 = 28 - 10\sqrt{3}\)
2. \((2 - 3\sqrt{3})^2 = 4 - 12\sqrt{3} + 27 = 31 - 12\sqrt{3}\)
3. \((2\sqrt{3} - 1)^2 = 12 - 4\sqrt{3} + 1 = 13 - 4\sqrt{3}\)

Now put it all together: \[ (28 - 10\sqrt{3}) + (31 - 12\sqrt{3}) - (13 - 4\sqrt{3}) \]

Combine like terms: \[ (28 + 31 - 13) + (-10\sqrt{3} - 12\sqrt{3} + 4\sqrt{3}) = 46 - 18\sqrt{3} \]

Answer: \( 46 - 18\sqrt{3} \)

г) Simplify \( \sqrt{28 - 10\sqrt{3}} + \sqrt{31 - 12\sqrt{3}} - \sqrt{13 - 4\sqrt{3}} \)

For each square root, we need to simplify:

1. \( \sqrt{28 - 10\sqrt{3}} \)

   • This can be expressed as \( \sqrt{(a - b)^2} \) where \( a^2 + b^2 = 28 \) and \( 2ab = 10\sqrt{3} \).
   • Solve \( ab = 5\sqrt{3} \).
   • Let \( a = 5 \) and \( b = \sqrt{3} \) results in: \[ (5^2 + (\sqrt{3})^2 = 25 + 3 = 28) \rightarrow \sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3} \]

2. \( \sqrt{31 - 12\sqrt{3}} \):

   • Similar structure gives \(a = 6\) and \( b = \sqrt{3}\)
   • Result: \(6 - \sqrt{3}\).

3. \( \sqrt{13 - 4\sqrt{3}} \):

   • Gives \(a = 4\) and \(b = \sqrt{3}\)
   • Result: \(4 - \sqrt{3}\).

Combine: \[ (5-\sqrt{3}) + (6-\sqrt{3}) - (4-\sqrt{3}) = 5 + 6 - 4 - \sqrt{3} - \sqrt{3} + \sqrt{3} = 7 - \sqrt{3} \]

Answer: \( 7 - \sqrt{3} \)

Feel free to ask more questions or request clarifications on any part!