Question

Evaluate the following using suitable identities: ${\left( {999} \right)^3}$

Answer 1

Hint:
We are asked in the question to Evaluate ${\left( {999} \right)^3}$ .
Since, we will split 999 as 1000-1. 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. 1000-1
Thus, solving further we will get the required answer.

Complete step by step solution:
We are asked in the question to Evaluate ${\left( {999} \right)^3}$ .
Since, we can split 999 as 1000-1.
Therefore, we can write ${\left( {999} \right)^3}$ as ${\left( {1000 - 1} \right)^3}$ .
 $ = {\left( {1000 - 1} \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( 1 \right)^3} - 3\left( {1000} \right)\left( 1 \right)\left( {1000 - 1} \right)$
 $ = 1000000000 - 1 - 3000 \times 999$
$=1000000000-1-2997000 \\
=997002999$

Hence, ${\left( {999} \right)^3} = 997002999$.

Note:
Here, students 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.
1) ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$
2) ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$
3) $\left( {{a^3} + {b^3}} \right) = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
4) $\left( {{a^3} - {b^3}} \right) = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$