Question 1204792
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Find the value of a if 7^a -7^a-5 = 117642√7
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<pre>
Your starting equation is this

    {{{7^a}}} - {{{7^(a-5)}}} = {{{117642*sqrt(7)}}}.


Factor left side by taking the factor {{{7^(a-5)}}}  out parentheses.

You will get an EQUIVALENT equation

     {{{7^(a-5)*(7^5-1)}}} = {{{117642*sqrt(7)}}}.


It is the same as 

    {{{7^(a-5)*16806}}} = {{{117642*sqrt(7)}}}.


Divide both sides by integer number 16806.

Notice that 117642 : 16806 = 7  with no remainder.

After dividing (canceling common factor 16806), you have

    {{{7^(a-5)}}} = {{{7*sqrt(7)}}}.


From it, you conclude that  a-5 = 1.5  (it is seen even by unarmed eye).

So, the <A>ANSWER</U> is : a = 5 + 1.5 = 6.5.
</pre>

Solved.


This solution is short and straightforward.