https://file.aiquickdraw.com/mathbot/file/1728146060765lpgs6js1.jpg

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

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### **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**.

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#### 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 **\( 4\sqrt{6} + 10 \)**.

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#### 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 **\( 9 - 5\sqrt{3} \)**.

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#### 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 **\( 25\sqrt{10} - 135 \)**.

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### **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.

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Would you like me to proceed with the remaining questions?

Solve the algebraic and square root expressions shown in the image for question 24.

Learn how to simplify square root and algebraic expressions such as (1+2√3)(1-2√3) and others using distributive properties and binomial expansion. Suitable for Grades 9-10 students studying algebra.

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