Question

Evaluate the given expression: ${\left( {1002} \right)^3}$ .

Answer 1

Hint: We are asked in the question to Evaluate ${\left( {1002} \right)^3}$ .
Since, we will split 1002 as 1000+2. After that, by applying the formula ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$ on the above equation i.e. 1000+2
Thus, solving further we will get the required answer.

Complete step-by-step answer:
We are asked in the question to Evaluate ${\left( {1002} \right)^3}$ .
Since, we can split 1002 as 1000+2.
Therefore, we can write ${\left( {1002} \right)^3}$ as ${\left( {1000 + 2} \right)^3}$ .
 $ = {\left( {1000 + 2} \right)^3}$
Now, applying the formula ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$ on the above equation, we get,
 $ = {\left( {1000} \right)^3} + {\left( 2 \right)^3} + 3\left( {1000} \right)\left( 2 \right)\left( {1000 + 2} \right)$
 $ = 1000000000 + 8 + 6000 \times 1002$
 $ = 1000000000 + 8 + 6012000$
 $ = 1006012008$
Hence, ${\left( {1002} \right)^3} = 1006012008$ .

Note: Here, student get confused between the ${\left( {a + b} \right)^3}$ and $\left( {{a^3} + {b^3}} \right)$ . So, apply the correct formula to get the correct required answer.
 ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$
 ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$
 $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
 $\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$