If a - b = 4 and ab = 21, find the value of $a^{3} - b^{3} .$

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Solution

a- b = 4, ab = 21

Cubing both sides,

$\Rightarrow (a - b)^{3} = (4)^{3} \Rightarrow a^{3} - b^{3} - 3 a b (a - b) = 64 \Rightarrow a^{3} - b^{3} - 3 \times 21 \times 4 = 64 \Rightarrow a^{3} - b^{3} - 252 = 64 \Rightarrow a^{3} - b^{3} = 64 + 252 = 316 ∴ a^{3} - b^{3} = 316$

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