`(sqrt(2)+sqrt(7-2sqrt(10)))` is equal to

Options

A. `sqrt(2)`
B. `sqrt(7)`
C. `sqrt(5)`
D. `2sqrt(5)`

Answer

To solve the expression \( \sqrt{2} + \sqrt{7 - 2\sqrt{10}} \), we can simplify the term inside the square root. Here’s a step-by-step breakdown of the solution:

### Step 1: Simplify the expression inside the square root
We start with the expression:
\[ \sqrt{7 - 2\sqrt{10}} \]

### Step 2: Recognize the form of a perfect square
We want to express \( 7 - 2\sqrt{10} \) as a perfect square. We look for two numbers \( a \) and \( b \) such that:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
We need \( a^2 + b^2 = 7 \) and \( 2ab = 2\sqrt{10} \).

### Step 3: Solve for \( a \) and \( b \)
From \( 2ab = 2\sqrt{10} \), we can simplify to:
\[ ab = \sqrt{10} \]
Let’s assume \( a = \sqrt{5} \) and \( b = \sqrt{2} \). Then:
\[ ab = \sqrt{5} \cdot \sqrt{2} = \sqrt{10} \]
Now we check \( a^2 + b^2 \):
\[ a^2 + b^2 = (\sqrt{5})^2 + (\sqrt{2})^2 = 5 + 2 = 7 \]
Both conditions are satisfied.

### Step 4: Rewrite the expression
Thus, we can write:
\[ 7 - 2\sqrt{10} = (\sqrt{5} - \sqrt{2})^2 \]
So we have:
\[ \sqrt{7 - 2\sqrt{10}} = \sqrt{(\sqrt{5} - \sqrt{2})^2} = \sqrt{5} - \sqrt{2} \]

### Step 5: Substitute back into the original expression
Now we substitute back into the original expression:
\[ \sqrt{2} + \sqrt{7 - 2\sqrt{10}} = \sqrt{2} + (\sqrt{5} - \sqrt{2}) \]

### Step 6: Simplify the expression
The \( \sqrt{2} \) terms cancel out:
\[ \sqrt{2} + \sqrt{5} - \sqrt{2} = \sqrt{5} \]

### Final Answer
Thus, the expression simplifies to:
\[ \sqrt{5} \]

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