Final answer:
When solving x² = 49, we find two solutions, x = 7 and x = -7, because squaring either positive or negative 7 yields 49. However, for the equation x³ = 8, the solution is uniquely x = 2 since the cube of a positive number remains positive, making x = -2 incorrect as (-2)³ equals -8.
Step-by-step explanation:
When we solve an equation like x² = 49, we get two solutions: x = 7 and x = -7. This is because both 7² and (-7)² equal 49. However, for the equation x³ = 8, the solution is only x = 2. This is because cube roots have only one real solution. The key distinction here is in how multiplication affects positive and negative numbers. When two negative numbers multiply, the result is positive, such as (-4) x (-3) = 12. But with cube roots, there is a single real solution because the sign of the result is the same as the sign of the original number. Therefore, since 8 is positive, its cube root must also be a positive number, namely 2. Thus, x = -2 does not satisfy the equation x³ = 8 because (-2)³ equals -8, not 8.