Question

Evaluate the following product without multiplication $999\times 999$.

Answer 1

Hint: In this problem we need to evaluate the given value without multiplication. For this we are going to use the algebraic formulas by converting the given number into a sum or difference of two simple nearest round off numbers. In the given expression we can observe that number $999$ two times. We can write the number $999$ as the difference between $1000$ and $1$ which are two simple integers. After substituting these values in the given expression, we can apply the exponential formula $x\times x={{x}^{2}}$ and there by algebraic formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ and simplify the value by using basic mathematical operation to get the required result.

Complete step by step answer:
Given expression is $999\times 999$.
In the above expression we can observe the number $999$ two times.
We are going to write the number $999$ as the difference between $1000$ and $1$ in the given expression, then we will get
$999\times 999=\left( 1000-1 \right)\times \left( 1000-1 \right)$
Applying the exponential formula $x\times x={{x}^{2}}$ in the above equation, then we will have
$999\times 999={{\left( 1000-1 \right)}^{2}}$
Applying the algebraic formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ in the above equation, then we will get
$999\times 999={{1000}^{2}}-2\left( 1000 \right)\left( 1 \right)+{{1}^{2}}$
Substituting the values ${{1000}^{2}}=1000000$, $2\left( 1000 \right)\left( 1 \right)=2000$, ${{1}^{2}}=1$ in the above equation and simplifying the equation by using basic mathematical operation, then we will have
$\begin{align}
  & 999\times 999=1000000-2000+1 \\
 & \Rightarrow 999\times 999=998000+1 \\
 & \Rightarrow 999\times 999=998001 \\
\end{align}$

Hence the value of $999\times 999$ is $998001$.

Note: In this problem we have the same number as two times in the given expression so we have used the exponential formula and after that algebraic formula. In some cases, we may have two different values in the expression then we can use the algebraic formula $\left( x+a \right)\left( x+b \right)={{x}^{2}}+\left( a+b \right)x+ab$.