The rule where raising a product to a power is the same as raising each element of the product to the power before multiplying makes sense, but I was wondering why the same is not true of a sum?
I accept that (a + b)^x =/= (a^x + b^x) and that you have to find the sum before raising it to a power, but I was wondering if someone could mathematically explain why this is the case?
I accept that (a + b)^x =/= (a^x + b^x) and that you have to find the sum before raising it to a power, but I was wondering if someone could mathematically explain why this is the case?
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This is due to the expansion that would be required by raising a sum to a certain power. As a basic example...
(a + b)² = a² + b² + 2ab ≠ a² + b²
While...
(ab)ⁿ = a*b*a*b...n times = (a times itself n times) * (b times itself n times) = aⁿbⁿ
In a more general sense, with sums, (a + b)ⁿ will generate a result that follows the binomial theorem.
(a + b)² = a² + b² + 2ab ≠ a² + b²
While...
(ab)ⁿ = a*b*a*b...n times = (a times itself n times) * (b times itself n times) = aⁿbⁿ
In a more general sense, with sums, (a + b)ⁿ will generate a result that follows the binomial theorem.