If $a - b = 3$ and $a b = 10$ , find $a^{3} - b^{3} =$ .

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Solution

Using the identity $(a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b)$

It is given that $a - b = 3$ and $a b = 10$ , therefore,

${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b) \Rightarrow {(3)}^{3} = a^{3} - b^{3} - (3 \times 10) (3) \Rightarrow 27 = a^{3} - b^{3} - (30 \times 3) \Rightarrow 27 = a^{3} - b^{3} - 90 \Rightarrow a^{3} - b^{3} = 27 + 90 \Rightarrow a^{3} - b^{3} = 117$

Hence, $a^{3} - b^{3} = 117$ .

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