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I see the uploaded image contains a set of mathematical problems, mostly dealing with simplifications involving square roots, rationalization, and solving algebraic equations. Let's break down the first few questions:

### Problem 1:
If \( \sqrt{5} = 2.2361 \), then find the value of \( \frac{\sqrt{5}-1}{\sqrt{5}+1} \).

**Solution:**
Using the given approximation \( \sqrt{5} = 2.2361 \):

\[
\frac{\sqrt{5}-1}{\sqrt{5}+1} = \frac{2.2361 - 1}{2.2361 + 1} = \frac{1.2361}{3.2361}
\]

Now, we perform the division:

\[
\frac{1.2361}{3.2361} \approx 0.3823
\]

So, the value is approximately \( 0.3823 \).

### Problem 2:
If \( \sqrt{2} = 1.4142 \), then find the value of \( \frac{\sqrt{2}+1}{\sqrt{2}-1} \).

**Solution:**
Substituting the value of \( \sqrt{2} \):

\[
\frac{\sqrt{2}+1}{\sqrt{2}-1} = \frac{1.4142 + 1}{1.4142 - 1} = \frac{2.4142}{0.4142}
\]

Now, divide:

\[
\frac{2.4142}{0.4142} \approx 5.83
\]

So, the value is approximately \( 5.83 \).

### Problem 3:
If \( \sqrt{3} = 1.732 \), then find the value of \( \frac{\sqrt{3}+1}{\sqrt{3}-1} \).

**Solution:**
Using the given approximation \( \sqrt{3} = 1.732 \):

\[
\frac{\sqrt{3}+1}{\sqrt{3}-1} = \frac{1.732 + 1}{1.732 - 1} = \frac{2.732}{0.732}
\]

Now, divide:

\[
\frac{2.732}{0.732} \approx 3.73
\]

So, the value is approximately \( 3.73 \).

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Would you like me to continue with the remaining problems or go into more detail on any of these steps?

Here are five follow-up questions based on this:

1. How would the result change in Problem 1 if \( \sqrt{5} \) were rounded differently?
2. Can you rationalize the expression \( \frac{\sqrt{5}-1}{\sqrt{5}+1} \) to avoid decimals?
3. Why does rationalization help when dealing with expressions like \( \frac{\sqrt{2}+1}{\sqrt{2}-1} \)?
4. How can you represent irrational numbers like \( \sqrt{2} \) more precisely using a fraction approximation?
5. How would you solve \( \frac{\sqrt{n}+1}{\sqrt{n}-1} \) for any \( n \)?

### Tip:
Rationalizing a denominator can simplify calculations by removing square roots from the bottom of a fraction, often leading to easier arithmetic or algebraic manipulation.

A series of mathematical problems focusing on square root simplifications, rationalization, and algebraic equations involving surds.

This detailed guide walks through a set of problems focused on simplifying square roots, rationalizing surds, and solving algebraic expressions involving square roots. Step-by-step solutions provided, suitable for students in Grades 9-12.

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