`sqrt(8-2sqrt(15))` is equal to :

Options

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

Answer

To solve the expression \( \sqrt{8 - 2\sqrt{15}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{8 - 2\sqrt{15}} \]

### Step 2: Identify the form of the expression
We notice that the expression inside the square root resembles the form \( a^2 - 2ab + b^2 \), which can be factored as \( (a - b)^2 \).

### Step 3: Express \( \sqrt{15} \)
We can express \( \sqrt{15} \) as \( \sqrt{5 \times 3} \). This helps us in identifying \( a \) and \( b \) in the expression.

### Step 4: Set up the equation
We want to find \( a \) and \( b \) such that:
\[ 8 = a^2 + b^2 \quad \text{and} \quad -2\sqrt{15} = -2ab \]
From the second equation, we can simplify it to:
\[ ab = \sqrt{15} \]

### Step 5: Choose values for \( a \) and \( b \)
Let's assume:
\[ a = \sqrt{5} \quad \text{and} \quad b = \sqrt{3} \]
Now we check:
\[ ab = \sqrt{5} \cdot \sqrt{3} = \sqrt{15} \]
This satisfies the condition for \( ab \).

### Step 6: Calculate \( a^2 + b^2 \)
Now we calculate \( a^2 + b^2 \):
\[ a^2 = (\sqrt{5})^2 = 5 \quad \text{and} \quad b^2 = (\sqrt{3})^2 = 3 \]
Thus,
\[ a^2 + b^2 = 5 + 3 = 8 \]
This satisfies the first condition.

### Step 7: Write the expression in squared form
Now we can rewrite the original expression:
\[ \sqrt{8 - 2\sqrt{15}} = \sqrt{(\sqrt{5} - \sqrt{3})^2} \]

### Step 8: Simplify the square root
Taking the square root gives us:
\[ \sqrt{8 - 2\sqrt{15}} = \sqrt{5} - \sqrt{3} \]

### Final Answer
Thus, the final answer is:
\[ \sqrt{8 - 2\sqrt{15}} = \sqrt{5} - \sqrt{3} \]

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