Question

(3i)/(-2-2i) in standard form show work

Answer 1

Answer:

The given complex number in standard form is

$\displaystyle{\boxed{\red{\sf\:-\:\dfrac{3}{4}\:-\:\dfrac{3i}{4}}}}$

Step-by-step-explanation:

We have given a complex number.

We have to express it in standard form.

The given complex number is

$\displaystyle{\sf\:\dfrac{3i}{-\:2\:-\:2i}}$

$\displaystyle{\implies\sf\:\dfrac{3i}{-\:2\:-\:2i}\:\times\:\dfrac{-\:2\:+\:2i}{-\:2\:+\:2i}}$

$\displaystyle{\implies\sf\:\dfrac{3i\:(\:-\:2\:+\:2i\:)}{(\:-\:2\:-\:2i\:)\:(\:-\:2\:+\:2i\:)}}$

$\displaystyle{\implies\sf\:\dfrac{-\:6i\:+\:6i^2}{2^2\:-\:(\:2i\:)^2}}$

$\displaystyle{\implies\sf\:\dfrac{-\:6i\:+\:6\:\times\:(\:-\:1\:)}{4\:-\:2^2\:\times\:i^2}\:\qquad\cdots[\:\because\:i^2\:=\:-\:1\:]}$

$\displaystyle{\implies\sf\:\dfrac{-\:6i\:-\:6}{4\:-\:4\:\times\:(\:-\:1\:)}}$

$\displaystyle{\implies\sf\:\dfrac{-\:6i\:-\:6}{4\:-\:(\:-\:4\:)}}$

$\displaystyle{\implies\sf\:\dfrac{-\:6i\:-\:6}{4\:+\:4}}$

$\displaystyle{\implies\sf\:\dfrac{-\:6i\:-\:6}{8}}$

$\displaystyle{\implies\sf\:-\:\dfrac{\cancel{6}}{\cancel{8}}\:-\:\dfrac{\cancel{6}\:i}{\cancel{8}}}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:-\:\dfrac{3}{4}\:-\:\dfrac{3i}{4}}}}}$

Answer 2

Question:-

$\sf\red{ \frac{3i}{ - 2 - 2i} } \: \sf{in \: standard \: form.}$

Given:-

• We have given a complex number.

To Find:-

• We have to express it in standard form.

Solution:-

The given complex number is

$\longmapsto\tt\sf \: \frac{3i}{ - 2 - 2i}$

• Standard form = a + ib

$= \sf \: \frac{3i}{ - 2 - 2i} = \frac{ - 3}{2 + 2i}$

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

$\sf = \frac{ - 6i + 6i {}^{2} }{2 {}^{2} - (2i) {}^{2} }$

$\sf = \frac{ - 6i + 6i {}^{2} }{4 + 4}$

$\sf = \frac{ - 6i - 6}{8}$

$\sf{i {}^{2} = 1}$

$\sf (2i) {}^{2} = - 4$

$\sf \frac{ - 6i - 6}{8} = \frac{ - 6}{8} - \frac{6i}{8} = \sf \frac{ - 3}{4} - \frac{3i}{4}$

$\sf a = \frac{ - 3}{4} ,b = \frac{ - 3i}{4}$

Answer:-

The given complex number in the standard form is

$\sf\huge\mathfrak\red{ \frac{ - 3}{4} - \frac{3i}{4} }$