How to interpret Weibull distribution comparing to Exponential distribution?

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I know that we can derive Weibull distribution comparing to Exponential distribution. The relation is $ Y = X^{\alpha} $. But how to interpret it? Should I think about it as about series of same rate incidents, which I can imagine as life time? What about distributions as $ Z = X_1^{\alpha_1} X_2^{\alpha_2} \dots X_n^{\alpha_n} $, can I imagine it as life time, but with different events with different rates? Each $ X_k $ is Exponential distribution with different $ \lambda_k $ rate. Does this distribution have a name? I would be very thankful for answer, it will help me to further exploration.