Question

a^b=c,b^c=a and c^a=b ,then prove that abc=a....How to solve this ??

Answer 1

\[a^b=c,b^c=a , c^a=b , prove ,that , abc=a\] how to solve?

Answer 2

\[a^b=c,b^c=a ,c^a=b\]
obvıously we see that\[abc=a^b b^c c^a\]

Answer 3

we need to fine abc =a .. How?

Answer 4

two questions: what a, b and c are?
and did u write everything correctly? (just to check)

Answer 5

especially check if question says abc=a or somethıng
and why dıd you rıght....

Answer 6

wrıte

Answer 7

if \[a ^{b }=c,b ^{c }=a,c ^{a }=b,\] then prove that abc=a

Answer 8

does the questıon say wether a,b,c are posıtıve ıntergers or non posıtıve or just constants

Answer 9

Real numbers, though.... a,b, and c ARE real numbers, right?

Answer 10

\[a ^{b }=c,b ^{c }=a,c ^{a }=b \\ \\ from \\ \\ b^c=a----> b=a^{\frac{1}{c}} \\ \\ sub ~into ~ equation ~3, \\ \\ \huge c=a^{\frac{1}{ac}} \\ \\ sub ~into ~ equation ~1, \\ \\ \huge a^b=a^{\frac{1}{ac}} \\ \\ \huge a^{abc}=a \\ \\ \huge abc=1\]
Am I missing something? :|

Answer 11

No :)

Answer 12

I also got that abc = 1 but how does that prove that a = abc?

Answer 13

yay :D

Answer 14

hey, @.Sam.

\[\huge a^{abc}=a\]does not necessarily mean abc = 1
it could be that
a=1

Answer 15

On second thought, scratch that :D

Answer 16

I think abc does equal 1 though. If you put each equation into log form and substitute, it comes out that abc = 1 also

Answer 17

The way I (just) saw it was
\[\large a^{abc} = a\]would imply either
abc = 1
or
a = 1
BUT
a = 1 would imply
c = 1
and
b = 1

implying
abc = 1 anyway
so... yeah, scratch the last one :D

Answer 18

But if a = 1, the proof is basically done (abc = a)
So... abc = 1... is tricky :)

Answer 19

So lame... :\, normally they'd asked abc=1, but this one's with some spinners

Answer 20

\[a^b=c \rightarrow \log_{a}c=b \]
\[b=c^a \rightarrow a^b=a ^{ac}\]
But
\[\log_{a}c=ca=c \iff a=1 \]
But
\[abc=1 ; a=1, \rightarrow abc=a\]

Answer 21

Compliments of @Hoa (who is a very smart guy)

Answer 22

@Mertsj thanks for your compliment, but pleaaaase, take the compliment out. It's totoooooooo shoooow offf :)

Answer 23

@Hoa No. The truth is the truth.