Question 1094495
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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 = <font color=red size=4>54</font> which is the answer.


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Side Note: exponents of 1/2 indicate square root. So for example {{{x^(1/2)^"" = sqrt(x)}}}
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