The imaginary part of
$${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$ can be :
Solution
$3 + 2\sqrt { - 54}$<br><br>
$= 9 - 6 + 2\sqrt { - 54}$<br><br>
$= 9 + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$<br><br>
$= {3^2} + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$<br><br>
$= {\left( {3 + \sqrt 6 i} \right)^2}$<br><br>
Similarly, $\left( {3 - 2\sqrt { - 54} } \right) = {\left( {3 - \sqrt 6 i} \right)^2}$<br><br>
$$ \therefore {\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$<br><br>
$$ = \pm \left( {3 + \sqrt 6 i} \right) - \left[ { \pm \left( {3 - \sqrt 6 i} \right)} \right]$$<br><br>
$= 6, - 6,2\sqrt 6 i, - 2\sqrt 6 i$<br><br>
$\therefore$ Possible imaginary parts are $2\sqrt 6 i, - 2\sqrt 6 i$
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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