What is the value of `sqrt(23+2sqrt(130))` ?

Options

A. A) `sqrt(15) + sqrt(8)`
B. B) `sqrt(21) + sqrt(2)`
C. C) `sqrt(13) + sqrt(10)`
D. D) `sqrt(13) + sqrt(10)`

Answer

To find the value of \( \sqrt{23 + 2\sqrt{130}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{23 + 2\sqrt{130}} \]

### Step 2: Identify components
We can express \( 23 \) and \( 2\sqrt{130} \) in a way that resembles the expansion of a square. Notice that:
\[ 23 = 13 + 10 \]
and
\[ 2\sqrt{130} = 2\sqrt{13 \times 10} \]

### Step 3: Recognize the square form
We can rewrite the expression as:
\[ \sqrt{( \sqrt{13} + \sqrt{10} )^2} \]
This is because:
\[ (\sqrt{13} + \sqrt{10})^2 = \sqrt{13}^2 + \sqrt{10}^2 + 2 \cdot \sqrt{13} \cdot \sqrt{10} \]
which simplifies to:
\[ 13 + 10 + 2\sqrt{130} = 23 + 2\sqrt{130} \]

### Step 4: Simplify the square root
Now we can simplify:
\[ \sqrt{( \sqrt{13} + \sqrt{10} )^2} = \sqrt{13} + \sqrt{10} \]

### Conclusion
Thus, the value of \( \sqrt{23 + 2\sqrt{130}} \) is:
\[ \sqrt{13} + \sqrt{10} \]