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11 March, 19:57

Why can't we use the rules for exponents when the bases are not common?

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  1. 11 March, 21:45
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    Ok, the rules of the exponent come from a logic construction.

    If we have x^n

    this means that n is multiplied by itself n times.

    If we decompose n into a + b, we have:

    x by itself a times, and then x by itself b times, and for how the product works, this is equivalent:

    if n = 5, a = 2 and b = 3

    x^5 = (x*x*x*x*x) 5 times-

    x^5 = x^ (2 + 3) = (x^2) * (x^3) = (x*x*) * (x*x*x) = x*x*x*x*x = x^5

    And the same for the other rules:

    (x^n) ^b = x^n*b and such.

    Obviusly, this only works when we have a common base.

    So the correct answer is that we constructed the exponential rules in a way that only can be used when we have a common base, and this happens because to construct them, we started with common bases.
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