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,
 $={\left(1000\right)}^3-{\left(1\right)}^3-3\left(1000\right)\left(1\right)\left(1000-1\right)$
 $=1000000000-1-3000\times 999$
$$\begin{gathered} =1000000000-1-2997000 \\ =997002999 \end{gathered}$$

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) $\left(a^3-b^3\right)=\left(a-b\right)\left(a^2+ab+b^2\right)$