Question

Evaluate the following using appropriate identity: \[{\left( {99} \right)^3}\]

Answer 1

Hint: As it is observed in the question, we have to use the identity to solve it. The Identity must be of the cube that is \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\]. Implement with the help of this identity.

Formula Used: Here, we use the formula \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\]

Complete step-by-step answer:
 To evaluate the cube of this number we should try to convert it into a format of 100-x. here x is a constant that we need to change according to the need of the problem.
In our case we have to convert the cube of 99 into this format
\[{\left( {99} \right)^3}\]
So, let’s change 99 to 100-1 as keeping x=1 will help us to solve this problem.
\[{\left( {100 - 1} \right)^3}\]
Here, \[a = 100\] and \[b = 1\]
Using the formula: \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\]
Substituting the values of a and b in the formula given above;
\[{\left( {100 - 1} \right)^3} = {100^3} - {1^3} - 3 * {100^2} * 1 + 3 * 100 * {1^2}\]
On simplifying the squares and the cubes of the equation we get,
\[ \Rightarrow 1000000 - 1 - 3 * 10000 * 1 + 3 * 100 * 1\]
On multiplying all the values in the equation;
\[ \Rightarrow 1000000 - 1 - 30000 + 300\]
Adding all the values in the equation:
\[ \Rightarrow 1000300 - 30001\]
On, subtracting the above equation:
We get, \[ \Rightarrow 970299\]
Hence, \[{\left( {99} \right)^3}\]\[ = 970299\]

Additional information:
We use the identity because it makes the question simpler and easy to solve as well as easy to understand the concept of solving cubes of large numbers with less effort. In Question like this we can use \[\left( {{{10}^n} \pm x} \right)\] , where n is the length of number and x is the number needed to be subtracted or added to make the resultant equal to the original number.\[\]


Note: In these type of questions, either we use positive identity or negative identity that is \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\] or \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}\]. So, carefully check which identity is best suited for the problem we are solving like for this one we used \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\].