What is the value of ` sqrt(8 - 2 sqrt(15))` ?

Options

A. `sqrt""3 + sqrt""2`
B. `sqrt""5 - sqrt""3`
C. `sqrt""5+ sqrt""3`
D. `sqrt""7 - sqrt""2`

Answer

To find the value of \( \sqrt{8 - 2\sqrt{15}} \), we can follow these steps:

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

### Step 2: Rewrite the expression
We want to express \( 8 - 2\sqrt{15} \) in the form of a perfect square. We can assume it can be written as:
\[ (\sqrt{a} - \sqrt{b})^2 \]
where \( a \) and \( b \) are to be determined.

### Step 3: Expand the perfect square
Expanding \( (\sqrt{a} - \sqrt{b})^2 \) gives:
\[ a + b - 2\sqrt{ab} \]
We need to match this with \( 8 - 2\sqrt{15} \).

### Step 4: Set up equations
From the expansion, we can equate:
1. \( a + b = 8 \)
2. \( -2\sqrt{ab} = -2\sqrt{15} \)

From the second equation, we can simplify:
\[ \sqrt{ab} = \sqrt{15} \implies ab = 15 \]

### Step 5: Solve the equations
Now we have a system of equations:
1. \( a + b = 8 \)
2. \( ab = 15 \)

Let’s denote \( a \) and \( b \) as the roots of the quadratic equation:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values we have:
\[ x^2 - 8x + 15 = 0 \]

### Step 6: Factor the quadratic
Now, we factor the quadratic:
\[ (x - 3)(x - 5) = 0 \]
Thus, the roots are:
\[ x = 3 \quad \text{and} \quad x = 5 \]
This means \( a = 5 \) and \( b = 3 \) (or vice versa).

### Step 7: Substitute back
Now substituting back, we have:
\[ \sqrt{8 - 2\sqrt{15}} = \sqrt{(\sqrt{5} - \sqrt{3})^2} \]

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

### Final Answer
Thus, the value of \( \sqrt{8 - 2\sqrt{15}} \) is:
\[ \sqrt{5} - \sqrt{3} \]