What is the square root of
`((sqrt(5)+2))/((sqrt(5)-2))` ?

Options

A. `sqrt(5)-sqrt(4)`
B. `sqrt(5)+2`
C. `5+sqrt(2)`
D. `7`

Answer

To find the square root of the expression \(\frac{\sqrt{5}+2}{\sqrt{5}-2}\), we will follow these steps:

### Step 1: Rationalize the Denominator
To simplify the expression, we need to rationalize the denominator. We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{5}+2\).

\[ \frac{\sqrt{5}+2}{\sqrt{5}-2} \times \frac{\sqrt{5}+2}{\sqrt{5}+2} \]

### Step 2: Apply the Difference of Squares Formula
Now, we can simplify the expression. The numerator becomes:

\[ (\sqrt{5}+2)(\sqrt{5}+2) = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 2 + 2^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5} \]

The denominator becomes:

\[ (\sqrt{5}-2)(\sqrt{5}+2) = (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1 \]

### Step 3: Simplify the Expression
Now we can simplify our expression:

\[ \frac{9 + 4\sqrt{5}}{1} = 9 + 4\sqrt{5} \]

### Step 4: Take the Square Root
Now, we need to find the square root of \(9 + 4\sqrt{5}\).

To find \(\sqrt{9 + 4\sqrt{5}}\), we can assume it can be expressed in the form \(\sqrt{a} + \sqrt{b}\).

Let's set:

\[ \sqrt{9 + 4\sqrt{5}} = \sqrt{a} + \sqrt{b} \]

Squaring both sides gives us:

\[ 9 + 4\sqrt{5} = a + b + 2\sqrt{ab} \]

From this, we can equate the rational and irrational parts:

1. \(a + b = 9\)
2. \(2\sqrt{ab} = 4\sqrt{5}\) → \(\sqrt{ab} = 2\sqrt{5}\) → \(ab = 20\)

### Step 5: Solve the System of Equations
Now we have a system of equations:

1. \(a + b = 9\)
2. \(ab = 20\)

Let \(a\) and \(b\) be the roots of the quadratic equation \(x^2 - (a+b)x + ab = 0\):

\[ x^2 - 9x + 20 = 0 \]

Using the quadratic formula:

\[ x = \frac{9 \pm \sqrt{9^2 - 4 \cdot 20}}{2} \]

Calculating the discriminant:

\[ 9^2 - 80 = 81 - 80 = 1 \]

So, we have:

\[ x = \frac{9 \pm 1}{2} \]

This gives us:

\[ x = 5 \quad \text{or} \quad x = 4 \]

Thus, \(a = 5\) and \(b = 4\) (or vice versa).

### Final Step: Write the Final Answer
Now we can write:

\[ \sqrt{9 + 4\sqrt{5}} = \sqrt{5} + \sqrt{4} = \sqrt{5} + 2 \]

Thus, the final answer is:

\[ \sqrt{\frac{\sqrt{5}+2}{\sqrt{5}-2}} = \sqrt{5} + 2 \]