To solve the problem step-by-step, let's start by using the provided information and perform the calculations needed:
1. We are given [tex]\(\sqrt{3} = 1.732\)[/tex].
2. First, find [tex]\(\sqrt{\sqrt{3}}\)[/tex]:
- Calculate the square root of [tex]\(\sqrt{3}\)[/tex], which is approximately 1.732.
- [tex]\(\sqrt{\sqrt{3}} \approx 1.316\)[/tex]
3. Next, determine [tex]\( \frac{1}{\sqrt{3}} \)[/tex]:
- Calculate the reciprocal of 1.732.
- [tex]\(\frac{1}{\sqrt{3}} \approx 0.577\)[/tex]
4. Finally, substitute these values and compute the expression [tex]\(\sqrt{\sqrt{3}} - \frac{1}{\sqrt{3}} + 1\)[/tex]:
- Plugging the values into the expression:
[tex]\[
\sqrt{\sqrt{3}} - \frac{1}{\sqrt{3}} + 1 \approx 1.316 - 0.577 + 1
\][/tex]
- Simplifying this:
[tex]\[
1.316 - 0.577 + 1 \approx 1.738
\][/tex]
Therefore, the expression [tex]\(\sqrt{\sqrt{3}} - \frac{1}{\sqrt{3}} + 1\)[/tex] evaluates to approximately 1.738.
So, the answer among the options given is:
[tex]\((1.3160547101089681, 0.5773672055427251, 1.738687504566243)\)[/tex]
Therefore, the closest value that matches is none of the multiple given options. Thus, there is either an error in the given options or another issue to address with these exact values.
However, note this step-by-step approach:
- Calculated [tex]\(\sqrt{\sqrt{3}}\)[/tex],
- Calculated [tex]\( \frac{1}{\sqrt{3}}\)[/tex], then added 1 and finally
- Evaluates to [tex]\(\approx 1.738\)[/tex]
So based on provided details, it is evident none from the list matches exactly.
