Question

Evaluate the following expression:
$\left( \sqrt{11}-\sqrt{7} \right)\left( \sqrt{11}+\sqrt{7} \right)$.

Answer 1

Hint: Here Expression is given in form of ( a - b )( a + b ) so use the formula \[\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}\] where $a=\sqrt{11}\ \And \ b=\sqrt{7}$.

Complete step-by-step answer:
We know that \[\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}\].
In the expression, $\left( \sqrt{11}-\sqrt{7} \right)\left( \sqrt{11}+\sqrt{7} \right)$, $a=\sqrt{11}\ \And \ b=\sqrt{7}$.
\[\begin{align}
  & \therefore \left( \sqrt{11}-\sqrt{7} \right)\left( \sqrt{11}+\sqrt{7} \right)={{\left( \sqrt{11} \right)}^{2}}-{{\left( \sqrt{7} \right)}^{2}} \\
 & =\sqrt{11}.\sqrt{11}-\sqrt{7}.\sqrt{7} \\
 & =11-7 \\
 & =4 \\
\end{align}\]
Therefore, the answer is 4.
Note: We can solve this expression by opening the brackets( using BODMAS rule ) as well.
$\begin{align}
  & \left( \sqrt{11}-\sqrt{7} \right)\left( \sqrt{11}+\sqrt{7} \right)=\sqrt{11}\left( \sqrt{11}+\sqrt{7} \right)-\sqrt{7}\left( \sqrt{11}+\sqrt{7} \right) \\
 & =\sqrt{11}\times \sqrt{11}+\sqrt{11}\times \sqrt{7}-\sqrt{7}\times \sqrt{11}-\sqrt{7}\times \sqrt{7} \\
 & =11+\sqrt{77}-\sqrt{77}-7 \\
 & =11-7 \\
 & =4 \\
\end{align}$