Question

Evaluate the following by using the identities:-
\[28\times 32\]

Answer 1

Hint: - In this type of questions, we try to find a number from which another number is one time added and one time subtracted and then these two quantities are multiplied with each other.

Complete step-by-step answer:
If the asked product can be modified like what is mentioned above, then we can use identity to evaluate the product in a very easy way rather than the actual long process of multiplication.
The identity that might be used in this question is as follows
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\cdot \left( a-b \right)\]

As mentioned in the question, we have to evaluate the product using the identities.
Now, for bringing the asked product into the required form that is \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\cdot \left( a-b \right)\] we can see that if 30 is subtracted by 2 and 30 is added with 2, then we get the desired numbers of which the product is to be evaluated which can be seen as following
 \[28\times 32=\left( 30-2 \right)\left( 30+2 \right)\]
Now, we can use the identity to evaluate the above product very easily as follows
\[\begin{align}
  & ={{30}^{2}}-{{2}^{2}} \\
 & =900-4 \\
 & =896 \\
\end{align}\]
Hence, the product of 28 and 32 is given as 896.

Note: -The students can make a mistake if they don’t know the following identity as it is very essential for evaluating the given product very easily and this being the only possible method to solve this question.
The identity which is used in this question is as follows
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\cdot \left( a-b \right)\]