I am an 6th grade student.And just learning the rules of exponents . Please don't close the question.An explanation would be appreciable and I'll be very great full if the question is answered.
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1I guess the rule you mean is: "$a^x = a^y$ if and only if $x=y$". In case $a<0$ it could happen that $a^x$ is not even defined (in the real numbers), like $a^{1/2}$. And even when defined, you have to rule out $a=-1$ just as you rule out $a=1$. – GEdgar Feb 01 '20 at 17:56
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If $a<0,$ then the expression $$a^x$$ doesn't always have a definite real value for an arbitrary real number $x.$ But we often want something like this -- and we always have at least one real value for $a^x$ whenever $a>0.$ We usually choose the positive of these as the meaning of $a^x,$ when there are more than one possibility. We rule out the case $a=1$ since then we would always have the same value for $a^x$ regardless what value $x$ assumes.
Allawonder
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Could you please give an example of a^x not having a real value for an arbitrary real number x? – user459284 Feb 02 '20 at 19:12
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Thanks.Could you also give an example of a^x =a^y,but x is not equal to y for a<1 ? – user459284 Feb 03 '20 at 18:08
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@user459284 Well, take $a=0$ and let $x,y$ be arbitrary positive numbers. – Allawonder Feb 03 '20 at 20:23
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Give an example where a is less than 0,a^x = a^y,but x is not equal to y. – user459284 Feb 04 '20 at 08:40
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@user459284 Take $a=-1,$ and let $x$ and $y$ be positive even integers with $x\ne y,$ say. – Allawonder Feb 05 '20 at 17:16
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The rule seems very interesting .Could you help with a deeper understanding ? – user459284 Feb 07 '20 at 15:27
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