Question

Evaluate the following given expression ${{\left( 999 \right)}^{3}}$
\[\begin{align}
  & \text{A}.\text{ 997}00\text{2999} \\
 & \text{B}.\text{ 997}00\text{299} \\
 & \text{C}.\text{ 998}00\text{1} \\
 & \text{D}.\text{ 97}00\text{2999} \\
\end{align}\]

Answer 1

Hint: To solve this question, we will assume a variable for 999 (say as x) then we will calculate ${{x}^{2}}$ as $x\times x$ and finally we will calculate ${{x}^{3}}$ by formula ${{x}^{3}}={{x}^{2}}\times x$
The value of ${{x}^{2}}$ is obtained as above and ${{x}^{3}}$ can be easily obtained as ${{x}^{2}}\times x={{\left( 999 \right)}^{2}}\times \left( 999 \right)$

Complete step-by-step answer:
Given the expression is ${{\left( 999 \right)}^{3}}$
Let us assume the value of x = 999, then we will calculate the value of \[{{x}^{3}}={{\left( 999 \right)}^{3}}\]
Let \[x=999\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)}\]
Then the value of \[{{x}^{2}}=x\times x\]
Substituting value of x as 999 in above equation we get:
\[\begin{align}
  & {{x}^{2}}=x\times x \\
 & {{x}^{2}}=999\times 999 \\
 & {{x}^{2}}=998001\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (ii)} \\
\end{align}\]
So, we have the value of ${{x}^{2}}$ as 998001
Now, to calculate the value of ${{x}^{3}}$ we will use the formula:
\[{{x}^{3}}={{x}^{2}}\times x\]
Now, we will substitute the value of ${{x}^{2}}=998001$ from equation (ii) and the value of x = 999 from equation (i) in above obtained expression.
\[\begin{align}
  & {{x}^{3}}={{x}^{2}}\times x \\
 & {{x}^{3}}=998001\times 999 \\
 & {{x}^{3}}=997002999 \\
\end{align}\]
Therefore, the value ${{x}^{3}}=997002999$
Therefore, the value of \[{{\left( 999 \right)}^{3}}=997002999\]

So, the correct answer is “Option D”.

Note: Another way to solve this question is by using formula of ${{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right)$
As the value $999=\left( 1000-1 \right)$
Cubing both sides and using identity ${{\left( 1000-1 \right)}^{3}}$ as ${{\left( a-b \right)}^{3}}$ stated above we get:
\[\begin{align}
  & {{\left( 999 \right)}^{3}}={{\left( 1000-1 \right)}^{3}}={{\left( 1000 \right)}^{3}}-{{\left( 1 \right)}^{3}}-3\times 1000\times 1\left( 1000-1 \right) \\
 & \Rightarrow 1000000000-1-3000\left( 999 \right) \\
 & \Rightarrow 1000000000-1-2997000 \\
 & \Rightarrow 1000000000-2997001 \\
 & \Rightarrow 997002999 \\
\end{align}\]
Hence, option D is the correct answer.