Question 202363
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if (125/27)^x = the cubic root of 0.6 then the value of x is . . . 
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        This problem can be solved by applying brute force,  as tutor  @Theo did it.


        But it can be solved elegantly,  if to look at the numbers and to think a bit.



<pre>
Notice that the ratio  {{{125/27}}}  in the left side is nothing else as  {{{(5/3)^3}}}.

In the right side,  0.6 = {{{6/10}}} = {{{3/5}}}.


This suggests that the left side is  {{{(5/3)^(3x)}}},  while the right side is  {{{(5/3)^(-1/3)}}}.


So, our equation is

    {{{(5/3)^(3x)}}} = {{{(5/3)^(-1/3)}}}.


This implies

    3x = {{{-1/3}}},  which gives  x = {{{-1/9}}}.


<U>ANSWER</U>.  The solution to this equation is  x = {{{-1/9}}}.
</pre>

Solved, and the exact solution is found.


Notice that  {{{-1/9}}} = -0.111111111111111...



Happy learning &nbsp;(&nbsp;!&nbsp;)


This problem is nice &nbsp;! &nbsp;&nbsp;&nbsp;&nbsp;It is &nbsp;{{{highlight(highlight(specially))}}} &nbsp;designed so that you would get the solution in &nbsp;{{{highlight(highlight(PRECISELY))}}} &nbsp;{{{highlight(highlight(this))}}} &nbsp;form 

and would get the taste of mathematical discovery.