Question

find the multiplicative inverse of
root 5+2i/ 1-2i​

Answer 1

Step-by-step explanation:

Given:

$z = \frac{ \sqrt{5} + 2i}{1 - 2i}$

To find: Multiplicative Inverse of z

Solution:

We know that multiplicative inverse of a/b is b/a

So,

multiplicative inverse of z is 1/z

$= \frac{1 - 2i}{ \sqrt{5} + 2i }$

Multiply and divide by complex conjugate of denominator

$= \frac{1 - 2i}{ \sqrt{5} + 2i} \times \frac{ \sqrt{5} - 2i }{ \sqrt{5} - 2i } \\ \\ = \frac{(1 - 2i)( \sqrt{5} - 2i)}{( \sqrt{5})^{2} + ( {2)}^{2} } \\ \\ = \frac{ \sqrt{5} - 2i - 2 \sqrt{5}i - 4 }{5 + 4} \\ \\ \because \: {i}^{2} = - 1 \\ \\ = \frac{ \sqrt{5} - 4 - 2i (1 + \sqrt{5} )}{9} \\ \\ = \frac{ \sqrt{5} - 4}{9} -i\frac{ 2(1 + \sqrt{5} )}{9}$

Thus,

$\bold{\frac{ \sqrt{5} - 4}{9} -i\frac{ 2(1 + \sqrt{5} )}{9} ) }$

is the multiplicative inverse of Z.

Hope it helps you.

To learn more on brainly:

1)Find the multiplicative inverse of 1/(4-3i)

https://brainly.in/question/5881172

2)Solve the given quadratic equation:

x² + 3ix + 10 = 0

https://brainly.in/question/7853683

Answer 2

Step-by-step explanation:

Step-by-step explanation:

Given:

\begin{gathered}z = \frac{ \sqrt{5} + 2i}{1 - 2i} \\\end{gathered}

z=

1−2i

5

+2i

To find: Multiplicative Inverse of z

Solution:

We know that multiplicative inverse of a/b is b/a

So,

multiplicative inverse of z is 1/z

\begin{gathered}= \frac{1 - 2i}{ \sqrt{5} + 2i } \\ \\\end{gathered}

=

5

+2i

1−2i

Multiply and divide by complex conjugate of denominator

\begin{gathered}= \frac{1 - 2i}{ \sqrt{5} + 2i} \times \frac{ \sqrt{5} - 2i }{ \sqrt{5} - 2i } \\ \\ = \frac{(1 - 2i)( \sqrt{5} - 2i)}{( \sqrt{5})^{2} + ( {2)}^{2} } \\ \\ = \frac{ \sqrt{5} - 2i - 2 \sqrt{5}i - 4 }{5 + 4} \\ \\ \because \: {i}^{2} = - 1 \\ \\ = \frac{ \sqrt{5} - 4 - 2i (1 + \sqrt{5} )}{9} \\ \\ = \frac{ \sqrt{5} - 4}{9} -i\frac{ 2(1 + \sqrt{5} )}{9}\end{gathered}

=

5

+2i

1−2i

×

5

−2i

5

−2i

=

(

5

)

2

+(2)

2

(1−2i)(

5

−2i)

=

5+4

5

−2i−2

5

i−4

∵i

2

=−1

=

9

5

−4−2i(1+

5

)

=

9

5

−4

−i

9

2(1+

5

)

Thus,

\begin{gathered}\bold{\frac{ \sqrt{5} - 4}{9} -i\frac{ 2(1 + \sqrt{5} )}{9} ) }\\\end{gathered}

9

5

−4

−i

9

2(1+

5

)

)

is the multiplicative inverse of Z.

Hope it helps you.

To learn more on brainly:

1)Find the multiplicative inverse of 1/(4-3i)

https://brainly.in/question/5881172

2)Solve the given quadratic equation:

x² + 3ix + 10 = 0

https://brainly.in/question/7853683