The value of `(1)/(sqrt(17 + 12 sqrt(2)))` is closest to .........

Options

A. 1.4
B. 1.2
C. 0.14
D. 0.17

Answer

To solve the problem of finding the value of \( \frac{1}{\sqrt{17 + 12\sqrt{2}}} \), we will follow these steps:

### Step 1: Simplify the expression under the square root
We start with the expression \( 17 + 12\sqrt{2} \). To simplify this, we can express it in the form of a perfect square.

We can rewrite \( 17 \) as \( 8 + 9 \) and \( 12\sqrt{2} \) as \( 2 \cdot 6\sqrt{2} \).

### Step 2: Identify the perfect square
Notice that \( 8 \) can be expressed as \( (2\sqrt{2})^2 \) and \( 9 \) can be expressed as \( 3^2 \).

Thus, we can rewrite the expression as:
\[ 17 + 12\sqrt{2} = (2\sqrt{2})^2 + 2 \cdot (2\sqrt{2}) \cdot 3 + 3^2 \]
This matches the form \( a^2 + 2ab + b^2 = (a + b)^2 \) where \( a = 2\sqrt{2} \) and \( b = 3 \).

### Step 3: Write as a square
Therefore, we can express \( 17 + 12\sqrt{2} \) as:
\[ (2\sqrt{2} + 3)^2 \]

### Step 4: Take the square root
Now, taking the square root of both sides, we have:
\[ \sqrt{17 + 12\sqrt{2}} = 2\sqrt{2} + 3 \]

### Step 5: Substitute back into the original expression
Now, substituting this back into our original expression, we get:
\[ \frac{1}{\sqrt{17 + 12\sqrt{2}}} = \frac{1}{2\sqrt{2} + 3} \]

### Step 6: Rationalize the denominator
To simplify further, we can multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{1}{2\sqrt{2} + 3} \cdot \frac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} = \frac{3 - 2\sqrt{2}}{(2\sqrt{2} + 3)(3 - 2\sqrt{2})} \]

### Step 7: Calculate the denominator
Calculating the denominator:
\[ (2\sqrt{2})^2 - 3^2 = 8 - 9 = -1 \]
Thus, we have:
\[ \frac{3 - 2\sqrt{2}}{-1} = -3 + 2\sqrt{2} \]

### Step 8: Approximate the value
Now, we need to approximate \( -3 + 2\sqrt{2} \):
Since \( \sqrt{2} \approx 1.414 \):
\[ 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \]
Thus:
\[ -3 + 2\sqrt{2} \approx -3 + 2.828 \approx -0.172 \]
Taking the absolute value, we find that the value is approximately \( 0.172 \).

### Conclusion
Thus, the value of \( \frac{1}{\sqrt{17 + 12\sqrt{2}}} \) is closest to \( 0.17 \).