Question 25907
1. {{{(a^m)(a^n) = a^(m+n)}}}


eg {{{(a^3)(a^2) = a^5}}} means (axaxa) x (axa) which is axaxaxaxa --> {{{a^5}}}


2. {{{(a^m)/(a^n) = a^(m-n)}}}


eg {{{(a^5)(a^2) = a^3}}} means (axaxaxaxa) divided by (axa) which is axaxa --> {{{a^3}}}


3. {{{(a^m)^n = a^(mn)}}}


eg {{{(a^3)^2 = a^6}}} means (axaxa) x (axaxa) which is axaxaxaxaxa --> {{{a^6}}}


4. {{{a^0 = 1}}}


eg {{{(a^3)/(a^3) = 1}}} means (axaxa)/(axaxa) which is 1...but the powers in a division subtract, so we get 3-3 which is zero. The zero power of ANY number is 1.


5. Start with rule 1 and let n be -m. We therefore get {{{(a^m)(a^(-m)) = a^(m-m)}}}.


{{{(a^m)(a^(-m)) = a^0}}}
{{{(a^m)(a^(-m)) = 1}}}
{{{a^(-m) = 1/(a^m)}}}


eg {{{3^(-2)}}} means {{{1/(3^2)}}}



6. Start with rule 3 and let n be 1/m. We therefore get {{{(a^m)^(1/m) = a^(m*(1/m))}}}.


{{{(a^m)^(1/m) = a^1}}}
{{{(a^m)^(1/m) = a}}}


So, we raise a number "a" to power m and then raise to power 1/m and we get the answer "a".


So, the power 1/m means the "opposite of raising to power m", ie it is the mth root.


eg {{{25^(1/2)}}} is the square root of 25.
eg {{{32^(1/5)}}} is the fifth root of 32.


jon