Question

Evaluate the ${\left( {99} \right)^3}$ using suitable identities.

Answer 1

Hint:
We are asked in the question to evaluate ${\left( {99} \right)^3}$ using suitable identities.
Since, we will split 99 as 100 - 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. 100-1
Thus, solving further we will get the required answer.

Complete step by step solution:
It is given the question that we have to evaluate ${\left( {99} \right)^3}$ using suitable identities.
Since, we can split 99 as $\left( {100 - 1} \right)$ .
Therefore, ${\left( {99} \right)^3} = {\left( {100 - 1} \right)^3}$
Now, using formula ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$ 0n ${\left( {100 - 1} \right)^3}$ , we get,
 $ = {\left( {100 - 1} \right)^3}$
 $ = {100^3} - {1^3} - 3\left( {100} \right)\left( 1 \right)\left( {100 - 1} \right)$
 $ = 1000000 - 1 - 300 \times 99$
 $ = 1000000 - 1 - 300 \times 99$
 $ = 1000000 - 1 - 29700 \\
 =970299 $

Hence, ${\left( {99} \right)^3} = 970299$

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