SOLUTION: What is the value of a if 7^a-7^a-5= 117642√7?

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Question 1148139: What is the value of a if 7^a-7^a-5= 117642√7?
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Please think about when parentheses are needed when you type algebraic expressions.

Clearly you mean 7^a-7^(a-5), which is 7%5Ea-7%5E%28a-5%29, instead of 7^a-7^a-5, which is 7%5Ea-7%5Ea-5, which is just -5.

On the left, factor out a common factor of 7^(a-5):

7%5Ea-7%5E%28a-5%29+=+%287%5E%28a-5%29%29%287%5E5-1%29+=+%287%5E%28a-5%29%29%2A16806

On the right, using the prime factorization of 117642 or any other method, determine that 117642%2Asqrt%287%29+=+7%2A16806%2Asqrt%287%29

Then we have

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

from which we can obtain

7%5E%28a-5%29+=+7%2Asqrt%287%29+=+7%5E1.5

So

a-5+=+1.5
a+=+1.5%2B5+=+6.5

ANSWER: a = 6.5, or 13/2