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The expression \((z+4)(z-4)\) can be simplified using the difference of squares formula, which states that:
\[
(a+b)(a-b) = a^2 - b^2
\]
In this case, \(a = z\) and \(b = 4\). Therefore:
\[
(z+4)(z-4) = z^2 - 4^2 = z^2 - 16
\]
Now let's compare this result with the given options:
1. \(z^2 - 8z - 16\)
2. \(2z^2 - z^2 - 8\)
3. \(z(z) + 4z(z) - 4(4)\)
4. \(z^2 + 4z - 4z - 16\)
The expression equivalent to \(z^2 - 16\) is not directly listed, but let's analyze each option:
1. \(z^2 - 8z - 16\) does not match.
2. \(2z^2 - z^2 - 8\) simplifies to \(z^2 - 8\), which does not match.
3. \(z(z) + 4z(z) - 4(4)\) does not simplify to \(z^2 - 16\).
4. \(z^2 + 4z - 4z - 16\) simplifies to \(z^2 - 16\).
Thus, the expression that is equivalent to \((z+4)(z-4)\) is:
**\(z^2 - 16\) from option 4.**