To find the value of [tex]\(\frac{1}{\sqrt{18-\sqrt{32}}}\)[/tex], we need to follow these steps:
1. Evaluate the Inner Square Root: First, calculate the value of [tex]\(\sqrt{32}\)[/tex].
2. Substitute and Simplify the Expression: Use the value obtained in the first step to simplify the term under the square root in the denominator.
3. Evaluate the Outer Square Root: Calculate the square root of the simplified expression.
4. Calculate the Reciprocate: Finally, find the reciprocal of the value obtained in the third step.
Let's carry out each step in detail.
### Step 1: Evaluate the Inner Square Root
Calculate [tex]\(\sqrt{32}\)[/tex].
The value of [tex]\(\sqrt{32}\)[/tex] is approximately [tex]\(5.656854249492381\)[/tex].
### Step 2: Substitute and Simplify the Expression
Substitute [tex]\(\sqrt{32}\)[/tex] in the expression [tex]\(18 - \sqrt{32}\)[/tex]:
[tex]\[ 18 - \sqrt{32} \approx 18 - 5.656854249492381 \][/tex]
This simplifies to:
[tex]\[ 18 - 5.656854249492381 \approx 12.343145750507619 \][/tex]
### Step 3: Evaluate the Outer Square Root
Now, calculate the square root of [tex]\(12.343145750507619\)[/tex]:
[tex]\[ \sqrt{12.343145750507619} \approx 3.5132813366577436 \][/tex]
### Step 4: Calculate the Reciprocal
Finally, find the reciprocal of [tex]\(3.5132813366577436\)[/tex]:
[tex]\[ \frac{1}{3.5132813366577436} \approx 0.2846341935574452 \][/tex]
So, the value of [tex]\(\frac{1}{\sqrt{18-\sqrt{32}}}\)[/tex] is approximately [tex]\(0.2846341935574452\)[/tex].
