The square root of `7 + 2sqrt(10)` is

Options

A. `(sqrt(6) + 1)`
B. `(sqrt(3) + 2)`
C. `(sqrt(2) + sqrt(5))`
D. `(2 + sqrt(5))`

Answer

To find the square root of \( 7 + 2\sqrt{10} \), we can express it in a more manageable form. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{7 + 2\sqrt{10}} \]

### Step 2: Express 7 in a different form
We can express \( 7 \) as \( 2 + 5 \):
\[ \sqrt{(2 + 5) + 2\sqrt{10}} \]

### Step 3: Identify the components
Notice that \( \sqrt{10} \) can be rewritten as \( \sqrt{2 \cdot 5} \). Hence, we can rewrite the expression as:
\[ \sqrt{2 + 5 + 2\sqrt{2 \cdot 5}} \]

### Step 4: Recognize the perfect square
This expression resembles the expansion of a binomial square:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Here, let \( a = \sqrt{2} \) and \( b = \sqrt{5} \). Thus, we have:
\[ (\sqrt{2} + \sqrt{5})^2 = 2 + 5 + 2\sqrt{2 \cdot 5} = 7 + 2\sqrt{10} \]

### Step 5: Take the square root
Now we can take the square root of both sides:
\[ \sqrt{7 + 2\sqrt{10}} = \sqrt{(\sqrt{2} + \sqrt{5})^2} \]
This simplifies to:
\[ \sqrt{2} + \sqrt{5} \]

### Final Answer
Thus, the square root of \( 7 + 2\sqrt{10} \) is:
\[ \sqrt{2} + \sqrt{5} \]

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