Question

Find the imaginary part of

$((3 + 2 \sqrt{ - 54} ) ^{ \frac{1}{2} } - (3 - 2 \sqrt{ - 54)} ^{ \frac{1}{2} })$
​

Answer 1

Answer:

First, we will solve the $(3+\sqrt[2]{-54} )$

=>  $(3+\sqrt[2]{-54} )$

We can say √9 =3

So,

$=> | (\sqrt{9}+\sqrt[2]{-54}) | \\=>| \sqrt{9 + 216} |\\=> |\sqrt{225} |\\=> 15$

Now,

$=> (3+\sqrt[2]{-54})^{1/2} \\=> { (\sqrt{\frac{15 + 3}{2} } )+ (\sqrt{\frac{15 - 3}{2} } )\\\\ => ±(3+\sqrt{6} )\\\\So,\\=> (3+\sqrt[2]{-54})^{1/2} - (3-\sqrt[2]{-54})^{1/2}\\=> 6 or \sqrt[2]{6}$

The imaginary part is => ± 2√6

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Answer 2

TO FIND :-

the imaginary part of

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

SOLUTION: -

$\rightarrow (3+\sqrt[2]{-54} )(3+2−54)$

$\rightarrow (3+\sqrt[2]{-54} )(3+2−54)$

$\rightarrow$ √9 =3

So,

$| (\sqrt{9}+\sqrt[2]{-54}) |[tex] \\=| \sqrt{9 + 216} | [tex]\\= |\sqrt{225}| = 15$

Now,

$(3+\sqrt[2]{-54})^{1/2}$$》 { (\sqrt{\frac{15 + 3}{2} } )+ (\sqrt{\frac{15 - 3}{2} } )$$=> ±(3+\sqrt{6} )\\So,\\=> (3+\sqrt[2]{-54})^{1/2} - (3-\sqrt[2]{-54})^{1/2}\\=> 6 or \sqrt[2]{6}$

$》 imaginary \: part \: is =± 2√6$

HOPE IT HELPS :)