Evaluate $(1001)^{3}$ using suitable identities.

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Solution

Given expression can be written as
${(1001)}^{3} = {(1000 + 1)}^{3}$
We know that
${(a + b)}^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
$\Rightarrow$ $\begin{array}{rcl} {(1000 + 1)}^{3} & = & {(1000)}^{3} + {(1)}^{3} + 3 {(1000)}^{2} (1) + 3 (1000) {(1)}^{2} = 1000000000 + 3000000 + 3000 + 1 \\ = & 1003003001 \end{array}$
Hence the result is $\begin{array}{rcl} 1003003001 \end{array}$

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