Question

Evaluate the following using identities: \[{{48}^{2}}\].

Answer 1

Hint: From the question, it is clear that we have to solve this problem using identities. We know that \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\] and \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] are two important identities. So, by using \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\] and \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] identities this problem can be solved.

Complete step-by-step answer:
From the question, it is given that we have to evaluate \[{{48}^{2}}\] by using identities.
Let us assume that \[{{48}^{2}}\] is equal to x.
\[\Rightarrow x={{48}^{2}}\]
Let us assume this as equation (1).
\[\Rightarrow x={{48}^{2}}.....(1)\]
Let us write \[50-2=48\] in equation (1).
\[\Rightarrow x={{\left( 50-2 \right)}^{2}}\]
Let us assume this as equation (2).
\[\Rightarrow x={{\left( 50-2 \right)}^{2}}.....(2)\]
From the concept of identities, we can say that \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\] and \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\].
Let us use the identity \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\] in the equation (2) to get the final answer.
\[\begin{align}
  & \Rightarrow x={{50}^{2}}+{{2}^{2}}-2(50)(2) \\
 & \Rightarrow x=2500+4-200 \\
 & \Rightarrow x=2304 \\
\end{align}\]
Let us assume this as equation (3).
\[\Rightarrow x=2304.....(3)\]
From equation (3), we can say that the value of x is equal to 2304.
So, we can say that the value of \[{{48}^{2}}\] is equal to 2304.

Note: Students may have a misconception that \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] and \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]. But we know that \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] and \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]. If this small misconception is followed, then the final answer may get interrupted. So, students should avoid these mistakes while solving this problem. Students should also be aware of calculation mistakes while solving this problem. If a small mistake is made, then the final answer may get interrupted. So, students should avoid these mistakes while solving this problem such that the final answer can be obtained in a correct manner.