Final answer:
The correct solution to the equation z³ = 27 is z = 3, since 3 cubed equals 27. Other options presented do not satisfy the equation when cubed. It is important to check the solution by substituting it back into the original equation.
Step-by-step explanation:
To solve the equation z³ = 27, we search for a number that, when cubed, equals 27. To do this, we take the cube root of both sides of the equation. The cube root of 27 is 3, which gives us z = 3. Now, we know that cubing both positive and negative numbers can yield a positive result, but since we are dealing with real numbers and no negative sign is present on the right side of the equation, the solution is only positive 3.
The other options provided, such as z = 9, z = -9, and z = ±3, do not satisfy the original equation when cubed. Therefore, the correct answer is z = 3, as cubing 3 gives us 3³ = 3*3*3 = 27, which matches the initial equation.
It's also important to eliminate terms that do not make sense in the context of the problem. For instance, the information given such as dx²-y², 2Py, or other unrelated algebraic expressions do not have relevance to the question of solving z³ = 27.
Remember to always check the answer to see if it is reasonable by substituting the solution back into the original equation, ensuring that it holds true.