Subject: Is the law (a^m)^n = a^(mn) always true? I'm in training to be a math teacher--and I want to know the laws of exponents with all their gory exceptions and complications. One law for exponents states: (a^m)^n = a^(mn). However, is this always true? For instance, is it true that (a^2)^(1/2) = a^1 = a? It doesn't seem that it can be for ((-3)^2))^(1/2) since it can be evaluated one of two ways: ONE: evaluating the "inside" first yields 9^(1/2)= 3 OR TWO: evaluating the "outside" first (by multiplying the exponents and applying the law) yields (-3)^1= -3. So which of these answers is correct--and why? My calculator (a TI-89) indicates the first one (positive 3), which would be consistent with evaluating from left to right, as the order of operations requires when operations are on the same level. In addition, I've always read that SQRT(X^2) is the ABS(X), which makes sense and is consistent with my first answer. If this is right, this would mean the above law is NOT always true. However, I never see this exception mentioned in the laws of exponents that are presented to students. Generally, these laws are claimed to hold for all real number bases and exponents, with the exception of 0^0. In addition, I also note that if you evaluate ((-3)^(1/2))^2 you get the same answer either way you proceed: ONE: evaluating the inside first yields(i*SQRT(3))^2= -3 and TWO: evaluating the outside first yields, once again, (-3)^1= -3. My TI-89 gives me the same answer as well (negative) Yet were the above exponent law to apply, I could apply the commutative law and rearrange the exponents from 1/2 * 2 to 2 * 1/2 and thus regenerate the above problem. It seems to me that the particular "law" of exponents in question is really only valid for a > 0. Am I correct? Is there some gory list of rules of how to deal with these situations? Any help would be much appreciated.
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
)