Question

the imaginary part of (3+2√-54)^1/2 - (3-2√-54)^1/2 can be
a) √-6
b)√6
c)-2√6
d)6​

Answer 1

Answer:

The imaginary part of the given complex number can be 2 √6.

Step-by-step-explanation:

The given complex number is

$\displaystyle{\sf\:(\:3\:+\:2\:\sqrt{-\:54}\:)^{\frac{1}{2}}\:-\:(\:3\:-\:2\:\sqrt{-\:54}\:)^{\frac{1}{2}}}$

We have to find the imaginary part of this complex number.

Now,

$\displaystyle{\sf\:(\:3\:+\:2\:\sqrt{-\:54}\:)^{\frac{1}{2}}\:-\:(\:3\:-\:2\:\sqrt{-\:54}\:)^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:(\:3\:+\:2\:\sqrt{54}\:i\:)^{\frac{1}{2}}\:-\:(\:3\:-\:2\:\sqrt{54}\:i\:)^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:(\:3\:+\:2\:\sqrt{9\:\times\:6}\:i\:)^{\frac{1}{2}}\:-\:(\:3\:-\:2\:\sqrt{9\:\times\:6}\:i\:)^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:(\:3\:+\:2\:\times\:3\:\sqrt{6}\:i\:)^{\frac{1}{2}}\:-\:(\:3\:-\:2\:\times\:3\:\sqrt{6}\:i)^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:(\:9\:-\:6\:+\:2\:\times\:3\:\times\:\sqrt{6}\:i\:)^{\frac{1}{2}}\:-\:(\:9\:-\:6\:-\:2\:\times\:3\:\times\:\sqrt{6}\:i\:)^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:(\:3^2\:+\:(\:\sqrt{6}\:i\:)^2\:+\:2\:\times\:3\:\times\:\sqrt{6}\:i)^{\frac{1}{2}}\:-\:(\:3^2\:+\:(\:\sqrt{6}\:i\:)^2\:-\:2\:\times\:3\:\times\:\sqrt{6}\:i)^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:[\:(\:3\:+\:\sqrt{6}\:i\:)^2\:]^{\frac{1}{2}}\:-\:[\:(\:3\:-\:\sqrt{6}\:i\:)^2\:]^{\frac{1}{2}}}$

$\displaystyle{\implies\sf\:(\:3\:+\:\sqrt{6}\:i\:)^{\frac{1}{2}\:\times\:2}\:-\:(\:3\:-\:\sqrt{6}\:i\:)^{\frac{1}{2}\:\times\:2}}$

$\displaystyle{\implies\sf\:3\:+\:\sqrt{6}\:i\:-\:(\:3\:-\:\sqrt{6}\:i\:)}$

$\displaystyle{\implies\sf\:3\:+\:\sqrt{6}\:i\:-\:3\:+\:\sqrt{6}\:i}$

$\displaystyle{\implies\sf\:0\:+\:2\:\sqrt{6}\:i}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:Imaginary\:part\:=\:2\:\sqrt{6}\:}}}}$

Answer 2

Answer:

$\Huge{\textbf{\textsf{{\color{navy}{ᴀɴ}}{\purple{ᴡs}}{\pink{ᴇʀ}}{\color{pink}{:}}}}}$

b)√6

b)√6