To solve the given problem [tex]\( N^* = \sqrt{\sqrt{13} + 1} \cdot \sqrt{\sqrt{13} - 1} - 2(\sqrt{3} - 8) \)[/tex], let's break it down into smaller, manageable steps:
1. Calculate the first multiplicative term:
[tex]\[
\text{term}_1 = \sqrt{\sqrt{13} + 1}
\][/tex]
We find:
[tex]\[
\text{term}_1 = 2.146054816509585
\][/tex]
2. Calculate the second multiplicative term:
[tex]\[
\text{term}_2 = \sqrt{\sqrt{13} - 1}
\][/tex]
We find:
[tex]\[
\text{term}_2 = 1.6141720092555158
\][/tex]
3. Multiply the two terms together:
[tex]\[
\text{product\_terms} = \text{term}_1 \cdot \text{term}_2
\][/tex]
We calculate:
[tex]\[
\text{product\_terms} = 2.146054816509585 \cdot 1.6141720092555158 = 3.4641016151377544
\][/tex]
4. Calculate the second term:
[tex]\[
\text{second\_term} = 2(\sqrt{3} - 8)
\][/tex]
We find:
[tex]\[
\sqrt{3} \approx 1.732
\][/tex]
Then:
[tex]\[
\text{second\_term} = 2(1.732 - 8) = 2(-6.268) = -12.536
\][/tex]
5. Combine the results:
[tex]\[
N^* = \text{product\_terms} - \text{second\_term}
\][/tex]
Thus:
[tex]\[
N^* = 3.4641016151377544 - (-12.536) = 3.4641016151377544 + 12.536 = 16.0
\][/tex]
So, the final value is:
[tex]\[
N^* = 16.0
\][/tex]
