`sqrt(3+sqrt((5)))` is equal to

Options

A. `sqrt(5)+1`
B. `sqrt(3)+sqrt(2)`
C. `(sqrt(5)+1)/(sqrt((2)))`
D. `(sqrt(5)+1)(2)`

Answer

To solve the expression \( \sqrt{3 + \sqrt{5}} \), we will simplify it step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{3 + \sqrt{5}} \]

### Step 2: Assume a form for simplification
Let's assume that \( \sqrt{3 + \sqrt{5}} \) can be expressed in the form \( \sqrt{a} + \sqrt{b} \). Therefore, we have:
\[ \sqrt{3 + \sqrt{5}} = \sqrt{a} + \sqrt{b} \]

### Step 3: Square both sides
Squaring both sides gives us:
\[ 3 + \sqrt{5} = a + b + 2\sqrt{ab} \]

### Step 4: Equate rational and irrational parts
From the equation \( 3 + \sqrt{5} = a + b + 2\sqrt{ab} \), we can separate the rational and irrational parts:
1. \( a + b = 3 \) (rational part)
2. \( 2\sqrt{ab} = \sqrt{5} \) (irrational part)

### Step 5: Solve for \( ab \)
From the second equation, we can solve for \( ab \):
\[ \sqrt{ab} = \frac{\sqrt{5}}{2} \implies ab = \left(\frac{\sqrt{5}}{2}\right)^2 = \frac{5}{4} \]

### Step 6: Set up a system of equations
Now we have a system of equations:
1. \( a + b = 3 \)
2. \( ab = \frac{5}{4} \)

### Step 7: Use the quadratic formula
Let \( a \) and \( b \) be the roots of the quadratic equation \( x^2 - (a+b)x + ab = 0 \):
\[ x^2 - 3x + \frac{5}{4} = 0 \]

### Step 8: Calculate the discriminant
The discriminant \( D \) of this quadratic is:
\[ D = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot \frac{5}{4} = 9 - 5 = 4 \]

### Step 9: Find the roots
The roots are given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm 2}{2} = \frac{5}{2} \text{ or } \frac{1}{2} \]

### Step 10: Assign values to \( a \) and \( b \)
Thus, we can assign:
\[ a = \frac{5}{2}, \quad b = \frac{1}{2} \]
or vice versa.

### Step 11: Substitute back to find the original expression
Now substituting back, we have:
\[ \sqrt{3 + \sqrt{5}} = \sqrt{\frac{5}{2}} + \sqrt{\frac{1}{2}} = \frac{\sqrt{5}}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{\sqrt{5} + 1}{\sqrt{2}} \]

### Final Answer
Thus, the final answer is:
\[ \sqrt{3 + \sqrt{5}} = \frac{\sqrt{5} + 1}{\sqrt{2}} \]