Question

find the modulus of(3-2i)(2+3i)/(1+2i)(2-i)​

Answer 1

SOLUTION

TO DETERMINE

The modulus of

$\displaystyle \sf{ \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} }$

EVALUATION

Here the given complex number is

$\displaystyle \sf{ \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} }$

Hence the required modulus is

$\displaystyle \sf{ = \bigg| \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} \bigg| }$

$\displaystyle \sf{ = \frac{ | (3 - 2i) | \: | (2 + 3i) | }{ | (1 + 2i) | \: | (2 - i) | } }$

$\displaystyle \sf{ = \frac{ \sqrt{ {3}^{2} + {( - 2)}^{2} } \times \: \sqrt{ {2}^{2} + {3}^{2} } }{ \sqrt{ {1}^{2} + {2}^{2} } \: \times \: \sqrt{ {2}^{2} + {( - 1)}^{2} } } }$

$\displaystyle \sf{ = \frac{ \sqrt{ 9 + 4 } \times \: \sqrt{ 4 + 9 } }{ \sqrt{ 1 + 4} \: \times \: \sqrt{ 4 + 1}} }$

$\displaystyle \sf{ = \frac{ \sqrt{ 13 } \times \: \sqrt{ 13 } }{ \sqrt{ 5} \: \times \: \sqrt{ 5}} }$

$\displaystyle \sf{ = \frac{ 13 }{ 5} }$

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