Let's analyze each equation and the mistakes made by Daniel:
1. 25x^2 - 16 = 9
Daniel's mistake: He took the square root of each term separately, which is not correct.
Explanation: When we take the square root of an equation, we must take the square root of both sides of the equation, not just each term separately.
Correct solution:
25x^2 - 16 = 9
25x^2 = 25
x^2 = 1
x = ±1
Therefore, the correct solutions are x = 1 and x = -1.
2. z^3 - 2 = 6
Daniel's mistake: He correctly found the cube root of 8, but forgot to consider the negative cube root as well.
Explanation: When we take the cube root of an equation, we must consider both the positive and negative cube roots, since a cube can have both a positive and negative cube root.
Correct solution:
z^3 - 2 = 6
z^3 = 8
z = ∛8
z = 2 or -2
Therefore, the correct solutions are z = 2 and z = -2.