What a Chain

Algebra Level 4

$a^b=1,b^c=2,c^a=3$ If $a,b,c$ are rational,then $a+b+c=?$

2 solutions

Michael Mendrin
May 11, 2018

If $a^b=1$ , then $a$ could be either $1$ or $-1$ . We check out the possibilities, we can't be sure if any solution in reals exists for $a=-1$ .

If $a=1$ , then $c=3$ , so that $b=2^\frac{1}{3}$ which is not rational.

But if $a=-1$ , then $c=\frac{1}{3}$ , so that $b=2^3=8$ , which is rational, and $(-1)^8=1$ does check out.

In this case, $a+b+c=22/3=7.3333...$

You may want to clarify that the answer should be rounded to four decimal places.

Stefan Popescu - 3 years ago

...did you say four decimal places?

Michael Mendrin - 3 years ago

yes bc the answer is 7.3333

Stefan Popescu - 3 years ago

@Michael Mendrin Same way sir!! Umm Sir, there is a typo in the last line.........it should be 22/3 instead.......!!

Aaghaz Mahajan - 3 years ago

..force of habit... you know, $\frac{22}{7} \approx \pi$

Michael Mendrin - 3 years ago

Yeah.....that was what I noticed too.....!!

Aaghaz Mahajan - 3 years ago

admirable sir! though it's latest version of detecting the solar prominence...

nibedan mukherjee - 2 years, 3 months ago

Aparna Phadke
Feb 24, 2019

a = -1 b = 8 c = 1/3