What is the value of `sqrt(40+2sqrt(319))` ?

Options

A. `sqrt(15) + sqrt(8)`
B. `sqrt(21) + sqrt(2)`
C. `sqrt(22) + sqrt(11)`
D. `sqrt(29) + sqrt(11)`

Answer

To find the value of \( \sqrt{40 + 2\sqrt{319}} \), we can use the identity for the square of a binomial. The identity states that:

\[ (a + b)^2 = a^2 + b^2 + 2ab \]

### Step 1: Identify the structure
We want to express \( 40 + 2\sqrt{319} \) in the form \( (a + b)^2 \). Here, we can see that \( 40 \) is the constant term and \( 2\sqrt{319} \) is the term involving the square root.

### Step 2: Set up equations
Let’s assume:
\[ a^2 + b^2 = 40 \quad \text{and} \quad 2ab = 2\sqrt{319} \]

From the second equation, we can simplify to:
\[ ab = \sqrt{319} \]

### Step 3: Solve for \( a \) and \( b \)
Now we have two equations:
1. \( a^2 + b^2 = 40 \)
2. \( ab = \sqrt{319} \)

We can express \( b \) in terms of \( a \):
\[ b = \frac{\sqrt{319}}{a} \]

Substituting this into the first equation:
\[ a^2 + \left(\frac{\sqrt{319}}{a}\right)^2 = 40 \]

This simplifies to:
\[ a^2 + \frac{319}{a^2} = 40 \]

### Step 4: Multiply through by \( a^2 \)
Multiplying through by \( a^2 \) to eliminate the fraction gives:
\[ a^4 - 40a^2 + 319 = 0 \]

### Step 5: Let \( x = a^2 \)
Let \( x = a^2 \). Then we have:
\[ x^2 - 40x + 319 = 0 \]

### Step 6: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{40 \pm \sqrt{(-40)^2 - 4 \cdot 1 \cdot 319}}{2 \cdot 1} \]
\[ x = \frac{40 \pm \sqrt{1600 - 1276}}{2} \]
\[ x = \frac{40 \pm \sqrt{324}}{2} \]
\[ x = \frac{40 \pm 18}{2} \]

### Step 7: Calculate the roots
Calculating the two possible values for \( x \):
1. \( x = \frac{58}{2} = 29 \)
2. \( x = \frac{22}{2} = 11 \)

### Step 8: Find \( a \) and \( b \)
Thus, we have:
\[ a^2 = 29 \quad \text{and} \quad b^2 = 11 \]
This implies:
\[ a = \sqrt{29} \quad \text{and} \quad b = \sqrt{11} \]

### Step 9: Final expression
Now substituting back into our expression:
\[ \sqrt{40 + 2\sqrt{319}} = \sqrt{(\sqrt{29} + \sqrt{11})^2} \]
Thus, we have:
\[ \sqrt{40 + 2\sqrt{319}} = \sqrt{29} + \sqrt{11} \]

### Conclusion
The final value is:
\[ \sqrt{40 + 2\sqrt{319}} = \sqrt{29} + \sqrt{11} \]