`sqrt(10+sqrt((24))+sqrt((40))+sqrt((60)))` is equal to

Options

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

Answer

To solve the expression \( \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} \), we can follow these steps:

### Step 1: Rewrite the terms under the square root
We start by rewriting each term in the expression:

1. \( 10 = \sqrt{5^2} + \sqrt{3^2} + \sqrt{2^2} \)
2. \( \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} = 2\sqrt{2 \cdot 3} \)
3. \( \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} = 2\sqrt{5 \cdot 2} \)
4. \( \sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15} = 2\sqrt{3 \cdot 5} \)

Now, we can express \( \sqrt{24} \), \( \sqrt{40} \), and \( \sqrt{60} \) in terms of their prime factors.

### Step 2: Substitute the rewritten terms
Now we can substitute these values back into the original expression:

\[ \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} = \sqrt{10 + 2\sqrt{6} + 2\sqrt{10} + 2\sqrt{15}} \]

### Step 3: Combine the terms
Next, we can combine these terms under a single square root:

\[ = \sqrt{(\sqrt{5})^2 + (\sqrt{3})^2 + (\sqrt{2})^2 + 2(\sqrt{2})(\sqrt{3}) + 2(\sqrt{5})(\sqrt{2}) + 2(\sqrt{3})(\sqrt{5})} \]

### Step 4: Recognize the perfect square
The expression inside the square root can be recognized as a perfect square:

\[ = \sqrt{( \sqrt{5} + \sqrt{3} + \sqrt{2})^2} \]

### Step 5: Take the square root
Taking the square root of both sides gives us:

\[ = \sqrt{5} + \sqrt{3} + \sqrt{2} \]

### Final Answer
Thus, the final answer is:

\[ \sqrt{10 + \sqrt{24} + \sqrt{40} + \sqrt{60}} = \sqrt{5} + \sqrt{3} + \sqrt{2} \]