SOLUTION: Determine the value of (p+q+r); if p,q, and r are positive integers, p^q = 8, and r^(1/p) = 7 This is what I tried but my answer doesn't match the answer key. Please see below:

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Question 1094495: Determine the value of (p+q+r); if p,q, and r are positive integers, p^q = 8, and r^(1/p) = 7
This is what I tried but my answer doesn't match the answer key. Please see below:
p^q = 8 => 2^3 = 8
p = 2 and q = 3
Replacing the equation r^(1/p) = 7 with p gives me this:
r^(1/2) = 7
To make this equation true I assumed that the value of r should be 7:
7^(1/2) = 7
7 = 7
So, p+q+r = 2+3+7 = 12 but my answer key contains different answer. Please help.

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

You have p = 2 and q = 3 correct.
p^q = 2^3 = 8 is true

If r^(1/p) = 7, then r^(1/2) = 7 meaning that r = 49 (not r = 7)
You square both sides to get rid of the exponent of 1/2 as shown below

r^(1/2) = 7
[ r^(1/2) ] ^2 = 7^2
r^(1/2*2) = 49
r^(2/2) = 49
r^1 = 49
r = 49

So in summary:
p = 2, q = 3, r = 49

Making
p+q+r = 2+3+49 = 5+49 = 54 which is the answer.

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Side Note: exponents of 1/2 indicate square root. So for example