Let's solve the problems from the image step by step. I'll begin with question 24, and proceed from there.

Question 24: Simplify the given expressions

The task here is to simplify each expression.

a) $(1 + 2\sqrt{3})(1 - 2\sqrt{3})$

This is a product of binomials of the form $(a + b)(a - b) = a^2 - b^2$.

• Here $a = 1$ and $b = 2\sqrt{3}$, so we apply the formula: $= 1^2 - (2\sqrt{3})^2 = 1 - 4 \times 3 = 1 - 12 = -11$ So the result is -11.

b) $(2\sqrt{3} - \sqrt{2})(3\sqrt{2} + 2\sqrt{3})$

This is a product of binomials, so we apply the distributive property: $(2\sqrt{3})(3\sqrt{2}) + (2\sqrt{3})(2\sqrt{3}) + (-\sqrt{2})(3\sqrt{2}) + (-\sqrt{2})(2\sqrt{3})$

Simplifying each term: $= 6\sqrt{6} + 4 \times 3 + (-\sqrt{4}) + (-2\sqrt{6})$ $= 6\sqrt{6} + 12 - 2 - 2\sqrt{6}$ $= (6\sqrt{6} - 2\sqrt{6}) + (12 - 2) = 4\sqrt{6} + 10$

So the result is $4\sqrt{6} + 10$.

c) $(3 - \sqrt{3})(2 - \sqrt{3})$

Using distributive property: $= (3)(2) + (3)(-\sqrt{3}) + (-\sqrt{3})(2) + (-\sqrt{3})(-\sqrt{3})$ $= 6 - 3\sqrt{3} - 2\sqrt{3} + 3$ $= 9 - 5\sqrt{3}$

So the result is $9 - 5\sqrt{3}$.

d) $\left( 5\sqrt{2} - \sqrt{125} \right) \left( \sqrt{125} - \sqrt{2} \right) - \sqrt{250}$

Let's first simplify $\sqrt{125}$ and $\sqrt{250}$:

• $\sqrt{125} = 5\sqrt{5}$
• $\sqrt{250} = 5\sqrt{10}$

Now simplify the expression $(5\sqrt{2} - 5\sqrt{5})(5\sqrt{5} - \sqrt{2})$: $= (5\sqrt{2})(5\sqrt{5}) + (5\sqrt{2})(-\sqrt{2}) + (-5\sqrt{5})(5\sqrt{5}) + (-5\sqrt{5})(-\sqrt{2})$ $= 25\sqrt{10} - 10 - 25 \times 5 + 5\sqrt{10}$ $= 25\sqrt{10} - 10 - 125 + 5\sqrt{10}$ $= 30\sqrt{10} - 135$

Finally, subtract $\sqrt{250} = 5\sqrt{10}$: $30\sqrt{10} - 135 - 5\sqrt{10} = 25\sqrt{10} - 135$

So the result is $25\sqrt{10} - 135$.

Question 25: Simplify the given expressions

We continue simplifying the expressions.

a) $\left( \frac{3\sqrt{5} + 1}{3\sqrt{5} - 1} \right)$

This expression is in a simplified form. It represents a ratio and doesn't need further simplification unless rationalization is required, but here it seems to be left as is.

Would you like me to proceed with the remaining questions?