We start with the equation given:
[tex]\[
\frac{1}{z} = 7 - 4 \sqrt{3}
\][/tex]
We need to solve for [tex]\( z \)[/tex]. To do this, we take the reciprocal of both sides of the equation:
[tex]\[
z = \frac{1}{7 - 4 \sqrt{3}}
\][/tex]
Next, we'll rationalize the denominator. We multiply the numerator and the denominator by the conjugate of the denominator [tex]\( 7 + 4\sqrt{3} \)[/tex]:
[tex]\[
z = \frac{1}{7 - 4 \sqrt{3}} \cdot \frac{7 + 4 \sqrt{3}}{7 + 4 \sqrt{3}} = \frac{7 + 4\sqrt{3}}{(7 - 4\sqrt{3})(7 + 4\sqrt{3})}
\][/tex]
We then simplify the denominator using the difference of squares:
[tex]\[
(7 - 4 \sqrt{3})(7 + 4 \sqrt{3}) = 7^2 - (4 \sqrt{3})^2 = 49 - 48 = 1
\][/tex]
This simplifies to:
[tex]\[
z = 7 + 4 \sqrt{3}
\][/tex]
Now, we need to find [tex]\( z^2 \)[/tex]:
[tex]\[
z^2 = (7 + 4 \sqrt{3})^2
\][/tex]
Using the binomial expansion formula [tex]\( (a+b)^2 = a^2 + 2ab + b^2 \)[/tex], we calculate:
[tex]\[
(7 + 4 \sqrt{3})^2 = 7^2 + 2 \cdot 7 \cdot 4 \sqrt{3} + (4 \sqrt{3})^2
\][/tex]
Simplifying each term, we get:
[tex]\[
7^2 = 49
\][/tex]
[tex]\[
2 \cdot 7 \cdot 4 \sqrt{3} = 56 \sqrt{3}
\][/tex]
[tex]\[
(4 \sqrt{3})^2 = 16 \cdot 3 = 48
\][/tex]
Adding these results together:
[tex]\[
(7 + 4 \sqrt{3})^2 = 49 + 56 \sqrt{3} + 48 = 97 + 56 \sqrt{3}
\][/tex]
Thus,
[tex]\[
z^2 = 97 + 56 \sqrt{3}
\][/tex]
Finally, we are asked to find [tex]\( z^2 - z^2 \)[/tex]:
[tex]\[
z^2 - z^2 = (97 + 56 \sqrt{3}) - (97 + 56 \sqrt{3}) = 0
\][/tex]
So, the value of [tex]\( z^2 - z^2 \)[/tex] is:
[tex]\[
\boxed{0}
\][/tex]
