I am trying to evaluate the sum below by using replacement rule but it doesn't work. I got some errors saying "...is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing".
Could anyone explain why the error happened and how to fix it?
sol = Solve[{a + b + c == 3, a^2 + b^2 + c^2 == 29, a*b*c == 11}, {a, b, c}];
sum = a^5 /. sol[[1,1]] + b^5 /. sol[[1,2]] + c^5 /. sol[[1,3]]
why the error happened
Plus has higher precedence than ReplaceAll(see: Operator Input Forms >> Operator Precedence), so addition is performed before replacement.
{#, Precedence @ #} & /@ {Plus, Rule, ReplaceAll}
{{Plus, 310.}, {Rule, 120.}, {ReplaceAll, 110.}}
A simpler example that generates the same error is
a^5 /. sol[[1]] + 5
how to fix it
You can parenthesize the terms to make sure that ReplaceAll is performed before Plus:
sum = (a^5 /. sol[[1, 1]]) + (b^5 /. sol[[1, 2]]) + (c^5 /. sol[[1, 3]]);
N[sum]
4138. + 9.99201*10^-16 I
Alternatively, you can use the following which avoids precedence issues:
sum2 = Total[{a, b, c}^5] /. sol[[1]];
N[sum2]
4138. + 9.99201*10^-16 I
a/.a->2 +5 there is no problem at all?
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a /. a->2 +5 the same as a/. a-> (2 +5) and gives, as expected 7. The case in your question is more like rule = a -> 2; a /. rule + 5 (or, a /. (a->2) +5), and this form gives an error.
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