What is the value of `(sqrt(5)+2)div(sqrt(5)-2)` ?

Options

A. a. `9+4sqrt(5)`
B. b. `9-4sqrt(5)`
C. c. `7+4sqrt(5)`
D. d. `9+2sqrt(5)`

Answer

To solve the expression \((\sqrt{5} + 2) \div (\sqrt{5} - 2)\), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \frac{\sqrt{5} + 2}{\sqrt{5} - 2} \]

### Step 2: Apply the componendo and dividendo rule
According to the componendo and dividendo rule, we can rewrite the expression as:
\[ \frac{(\sqrt{5} + 2) + (\sqrt{5} + 2)}{(\sqrt{5} - 2) + (\sqrt{5} + 2)} \]
This simplifies to:
\[ \frac{2(\sqrt{5} + 2)}{2\sqrt{5}} \]

### Step 3: Simplify the expression
Now we can simplify the expression:
\[ \frac{2(\sqrt{5} + 2)}{2\sqrt{5}} = \frac{\sqrt{5} + 2}{\sqrt{5}} \]

### Step 4: Split the fraction
We can split the fraction into two parts:
\[ \frac{\sqrt{5}}{\sqrt{5}} + \frac{2}{\sqrt{5}} = 1 + \frac{2}{\sqrt{5}} \]

### Step 5: Rationalize the second term
To rationalize \(\frac{2}{\sqrt{5}}\), we multiply the numerator and the denominator by \(\sqrt{5}\):
\[ 1 + \frac{2\sqrt{5}}{5} \]

### Step 6: Combine the terms
Now we can combine the terms:
\[ 1 + \frac{2\sqrt{5}}{5} = \frac{5}{5} + \frac{2\sqrt{5}}{5} = \frac{5 + 2\sqrt{5}}{5} \]

### Final Answer
Thus, the value of \((\sqrt{5} + 2) \div (\sqrt{5} - 2)\) is:
\[ \frac{5 + 2\sqrt{5}}{5} \]

---