If a + b = 7 and a^3 + b^3 = 44, what is the value of the sum 1/a + 1/b? Express your answer as a common fraction.
a + b = 7,
a^3 + b^3 = 44, solve for a, b
a≈3.5 - 1.41 i and b≈3.5 + 1.41 i
a≈3.5 + 1.41 i ∧ b≈3.5 - 1.41 i
1 / [3.5 + 1.41 i] + 1 / [3.5 - 1.41 i] =70,000 / 142,381 = 0.4916386315589861.......etc.
+2666 \({1 \over a} + {1 \over b} = {b \over ab} + {a \over ab} = {a + b \over ab} = {7 \over ab} \)
\((a+b)^3 = a^3 + b^3 + 3a^2b+3ab^2 = 343\)
\(44 + 3a^2b+3ab^2 = 343\)
\(3a^2b+3ab^2 = 299\)
\(3ab(a+b) = 299\)
\(3ab \times 7 = 299\)
\(21ab = 299\)
\(ab = {299 \over 21}\)
\({7 \over {299 \over 21}} = {{147 \over 7} \over {299 \over 21}} = \color{brown}\boxed{147 \over 299}\)
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