Question 3: A rectangle has sides of length \(2+\sqrt{18}m\) and \(\left(5-\frac{4}{\sqrt{2}}\right)m\). Express the area of the rectangle in the form \(a+b\sqrt{2}\), where \(a\) and \(b\) are integers.
Solution:
\[
\begin{align*}
A&=Length\times Width\\
&=\left( 2+\sqrt{18}\right)\left( 5-\frac{4}{\sqrt{2}}\right)\\
&\small{=\left( 2+\sqrt{2\times 3^{2}}\right)\left( 5-\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\right)}\\
&=\left( 2+3\sqrt{2}\right)\left( 5-\frac{4\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)\\
&=\left( 2+3\sqrt{2}\right)\left( 5-\frac{4\sqrt{2}}{2}\right)\\
&=\left( 2+3\sqrt{2}\right)\left( 5-2\sqrt{2}\right)\\
&\small{=2\left( 5-2\sqrt{2}\right) +3\sqrt{2}\left( 5-2\sqrt{2}\right)}\\
&=10-4\sqrt{2} +15\sqrt{2} -6\left(\sqrt{2}\right)^{2}\\
&=10+11\sqrt{2} -6\times 2\\
&=10+11\sqrt{2} -12\\
&=-2+11\sqrt{2} \ m^{2}\\
\end{align*}
\]