Final answer:
Hannah performed operations that are inverses of each other: cubing a number and then taking the cube root. These operations should return the original value, which in this case is 21. Hence, Hannah started with the number 21.
Step-by-step explanation:
Hannah cubes a number and then takes the cube root of the result, ending up with 21. We can use the mathematical operation of cubing and the reverse operation, the cube root, to find the original number Hannah started with. The process of cubing involves raising a number to the power of 3, which is mathematically the same as multiplying the number by itself three times. The cube root operation is the inverse of this, finding a number that when cubed gives the original value.
Since the result after these two operations is 21, we can assume that the number Hannah started with is also 21. This is because the operations of cubing a number and then taking the cube root should return the initial value. Mathematically, if x is the number Hannah started with, she performed the operations x³ followed by ∛x³, which results in x. In Hannah's example, since the final result is 21, we conclude x = 21.
Therefore, Hannah originally started with the number 21.