The imaginary part of (3+2-54)^1/2-(3-2-54)^1/2 can be:

Question

The imaginary part of $(3+2\sqrt{-54})^{1/2}-(3-2\sqrt{-54})^{1/2}$ can be:

Accepted Answer

$$3 + 2\sqrt { - 54} $$

$$ = 9 - 6 + 2\sqrt { - 54} $$

$$ = 9 + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$$

$$ = {3^2} + {\left( {\sqrt 6 i} \right)^2} + 2.3.\sqrt 6 i$$

$$ = {\left( {3 + \sqrt 6 i} \right)^2}$$

Similarly, $$\left( {3 - 2\sqrt { - 54} } \right) = {\left( {3 - \sqrt 6 i} \right)^2}$$

$$ \therefore {\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$

$$ =  \pm \left( {3 + \sqrt 6 i} \right) - \left[ { \pm \left( {3 - \sqrt 6 i} \right)} \right]$$

$$ = 6, - 6,2\sqrt 6 i, - 2\sqrt 6 i$$

$$ \therefore $$ Possible imaginary parts are $$2\sqrt 6 i, - 2\sqrt 6 i$$

Solutions · 1