To solve the inequality \( z^5 \neq 7 \), we first need to find the value of \( z \) that would make both sides equal, i.e., \( z^5 = 7 \).
To find this value, we take the fifth root of both sides:
\[ z = 7^{1/5} \]
Calculating \( 7^{1/5} \) numerically, it is approximately \( 1.93 \).
To identify which of the provided options \( 30, 35, 40, \) or \( 45 \) is not part of the solution set, we can check which of these values, when raised to the power of 5, equals 7.
However, a simpler approach is to realize that we need \( z^5 \) to equal 7 for the number to be excluded from the solution set. Since none of the options \( 30, 35, 40, \) or \( 45 \) are \( 1.93 \), we can check which of these values when divided by 5 goes closest to 7.
Calculating the fifth power of each option:
- \( z = 30 \): \[ 30^5 = 24300000 \]
- \( z = 35 \): \[ 35^5 = 52521875 \]
- \( z = 40 \): \[ 40^5 = 102400000 \]
- \( z = 45 \): \[ 45^5 = 183228225 \]
All of these outcomes are clearly greater than 7.
Thus, \( 7^{1/5} \approx 1.93 \) is far below any of these numbers. Therefore, no number will make \( z^5 = 7 \) among the options 30, 35, 40, or 45.
Thus, the conclusion is that:
None of the given options (30, 35, 40, 45) are part of the solution set. All are valid choices, but a specific answer should focus on one option for clarity; selecting any would be correct.
If you are looking for a number that is clearly not part of the solution set:
- Choose 30 or any other, as they all are indeed valid and above that threshold of equality with \( z^5 = 7 \).
However, since you wanted to identify one for a single-point answer, 30 can be a response just to denote choice.