SOLUTION: Find the value of a if 7^a -7^a-5 = 117642√7

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Question 1204792: Find the value of a if 7^a -7^a-5 = 117642√7
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll assume the equation is 7%5Ea+-+7%5E%28a-5%29+=+117642%2Asqrt%287%29

It can be rewritten to 7%5Ea+-+7%5E%28-5%29%2A7%5Ea+=+117642%2Asqrt%287%29 and then to x+-+7%5E%28-5%29%2Ax+=++117642%2Asqrt%287%29 where x+=+7%5Ea

Let's solve for x.
x+-+7%5E%28-5%29%2Ax+=++117642%2Asqrt%287%29

x%281+-+7%5E%28-5%29%29+=++117642%2Asqrt%287%29

x%281+-+1%2F%287%5E5%29%29+=++117642%2Asqrt%287%29

x%281+-+1%2F16807%29+=++117642%2Asqrt%287%29

x%2816807%2F16807+-+1%2F16807%29+=++117642%2Asqrt%287%29

x%28%2816807-1%29%2F16807%29+=++117642%2Asqrt%287%29

x%2816806%2F16807%29+=++117642%2Asqrt%287%29

x+=++%2816807%2F16806%29%2A117642%2Asqrt%287%29

x+=++117649%2Asqrt%287%29

Notice how 7^6 = 117649
We can rewrite the right hand side as a power of 7.

x+=+117649%2Asqrt%287%29

x+=+7%5E6%2A7%5E%281%2F2%29%5E%22%22

x+=+7%5E%286%2B1%2F2%29%5E%22%22

x+=+7%5E%2813%2F2%29%5E%22%22

7%5Ea+=+7%5E%2813%2F2%29%5E%22%22 plug in x = 7^a

a+=+13%2F2 The bases are both 7, so the exponents must be the same.

Answer by ikleyn(52785) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the value of a if 7^a -7^a-5 = 117642√7
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Your starting equation is this

    7%5Ea - 7%5E%28a-5%29 = 117642%2Asqrt%287%29.


Factor left side by taking the factor 7%5E%28a-5%29  out parentheses.

You will get an EQUIVALENT equation

     7%5E%28a-5%29%2A%287%5E5-1%29 = 117642%2Asqrt%287%29.


It is the same as 

    7%5E%28a-5%29%2A16806 = 117642%2Asqrt%287%29.


Divide both sides by integer number 16806.

Notice that 117642 : 16806 = 7  with no remainder.

After dividing (canceling common factor 16806), you have

    7%5E%28a-5%29 = 7%2Asqrt%287%29.


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

So, the ANSWER is : a = 5 + 1.5 = 6.5.

Solved.

This solution is short and straightforward.