In this video I go over the general exponent law with base a: (a·b)^x = a^xb^x. In proving it I use the laws of exponents with base e as well as the definition a^x = e^{x·ln a} which I covered earlier. Also I use the logarithm law ln(a·b) = ln a + ln b.

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Laws of Exponents – (ab)^x = a^x b^x

General Laws of Exponents

If x and y are real numbers and a > 0, then:

1. a^x+y = a^xa^y
2. a^x-y = a^x/a^y
3. (a^x)^y = a^xy
4. (ab)^x = a^xb^x

Recall from my previous videos:

1. e^x+y = e^xe^y
2. e^x-y = e^x/e^y
3. (e^x)^r = e^rx

Also Recall Definition: a^x = e^xlna
Also Recall: ln(ab) = ln a + ln b

Proof: