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So why weight should be multiplied with input?

Yes I know, weight is intended for tuning the connection strength of input that will affect output so that's will be useful for learning (CMIIW).

But why the input should be MULTIPLIED by weight? Why doesn't use EXPONENT operation?

like this,

y_hat = x_1^w_1 + x_2^w_2 + bias
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    $\begingroup$ A similar question is asked here: quora.com/… it's not always the case that multiplcation is used, but using exponential would case too large a number very quickly, see: towardsdatascience.com/… $\endgroup$
    – Rob
    Sep 14, 2022 at 0:18

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You could try to use it as an exponent, and it will still learn... However, you will almost certainly end up with numeric instability very quickly with values either exploding or vanishing.

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