find modulus of (1+3i)(2-5i)/ (2- root 6i) (-3+ 2 root 5i)
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Please find below the solution to the asked query:
$$\begin{gathered} \Rightarrow \frac{\left(1+3\mathrm{i}\right)\left(2-5\mathrm{i}\right)}{\left(2-\sqrt{6}\mathrm{i}\right)\left(-3+2\sqrt{5}\mathrm{i}\right)} \\ \Rightarrow \frac{2-5\mathrm{i}+6\mathrm{i}+15}{-6+4\sqrt{5}\mathrm{i}+3\sqrt{6}\mathrm{i}+2\sqrt{30}} \\ \Rightarrow \frac{17+\mathrm{i}}{\left(-6+2\sqrt{30}\right)+\left(4\sqrt{5}+3\sqrt{6}\right)\mathrm{i}} \mathrm{because} \sqrt{30}=5.4 \mathrm{and} \sqrt{5}=2.23 \mathrm{and} \sqrt{6}=2.4 \\ \mathrm{So} \\ \Rightarrow \frac{17+\mathrm{i}}{\left(-6+2\left(5.4\right)\right)+\left(4\left(2.23\right)+3\left(2.4\right)\right)\mathrm{i}} \\ \Rightarrow \frac{17+\mathrm{i}}{\left(4.8\right)+\left(16.12\right)\mathrm{i}} \\ \mathrm{Now} \mathrm{multiply} \mathrm{with} \mathrm{Denomenator} \mathrm{conjugate} \left[\left(4.8\right)-\left(16.12\right)\mathrm{i}\right]\mathrm{in} \mathrm{both} \mathrm{numerator} \mathrm{and} \mathrm{denomenator} \\ \Rightarrow \frac{\left(17+\mathrm{i}\right)\left[\left(4.8\right)-\left(16.12\right)\mathrm{i}\right]}{\left[\left(4.8\right)+\left(16.12\right)\mathrm{i}\right]\left[\left(4.8\right)-\left(16.12\right)\mathrm{i}\right]} \\ \mathrm{Apply} \mathrm{the} \mathrm{formula} {\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}-\mathrm{b}\right) \left(\mathrm{a}+\mathrm{b}\right) \\ \Rightarrow \frac{81.6-274.04\mathrm{i}+4.8\mathrm{i}+16.12}{23.04+259.85} \\ \Rightarrow \frac{97.72}{282.89}+\frac{\left(-269.24\right)\mathrm{i}}{282.89} \\ \Rightarrow 0.34-0.95\mathrm{i} \\ \mathrm{Because} \left|\mathrm{z}\right| =\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2} \mathrm{if} \mathrm{z}=\mathrm{a}+\mathrm{ib} \\ \mathrm{So} \\ \Rightarrow \sqrt{0.{34}^2+{\left(-0.95\right)}^2} \\ \Rightarrow \sqrt{0.1156+0.90} \\ \Rightarrow \sqrt{1.0181} \\ \Rightarrow 1.009 \end{gathered}$$


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