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 $3.73$.

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.