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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

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Exponent of Complex Numbers

» Play with the forms of the complex number

→ component form `a+ib`

→ polar form `r(cos theta + i sin theta)`

→ exponent form `re^(i theta)`

» Convert to the standard form of complex numbers `a+ib`

→ eg: `(a+ib)^(c+id)`: convert `(a+ib)` to polar form `re^(i theta)`

→ eg: `r^(id)` : convert `r` to `e^(ln r)`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

To find exponent and root of complex numbers, the rules of numerical expression is used to arrive at the coordinate form `a+ib`.

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

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Exponent of a complex number is explained.

Starting on learning "Exponent of a complex number". ;; In this page, Exponent of a complex number is explained.

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, Can the exponent `z_1^z_2` be written in `a+ib` form?

- No. Exponent to the power of complex number is not possible
- Yes. By converting to polar form `z_1 = r e^(i theta)`.

The answer is 'Yes. By converting to polar form `z_1 = r e^(i theta)'

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, exponent

`z_1^z_2`

`quad quad = (r_1e^(i theta_1) )^(a_2+ib_2)`

`quad quad = r_1^a_2 `

`quad quad quad quad xx r_1^(ib_2)`

`quad quad quad quad xx e^(i theta_1 a_2)`

`quad quad quad quad xx e^(i theta_1 i b_2)`

`quad quad = r_1^a_2 `

`quad quad quad quad xx e^(ib_2 ln r_1) (`

`quad quad quad quad xx e^(i theta_1 a_2)`

`quad quad quad quad xx e^(- theta_1 b_2)`

`quad quad = r_1^a_2 e^(- theta_1 b_2) `

`quad quad quad quad xx e^(i(b_2 ln r_1 + theta_1 a_2))`

The result is in the polar form and can be converted to coordinate form.

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, Can the root `root(z_2)(z_1)` be computed in `a+ib` form?

- No. root to a complex number is not possible
- Yes. By considering root as power of `(1/z_2)`

The answer is 'Yes. By considering root as power of `(1/z_2)`'

Given `z_1 = a_1+ib_1` and `z_2 = a_2+ib_2`, root

`root(z_2)(z_1)`

`quad quad = z_1^(1/z_2)`

`quad quad = z_1^(bar(z_2)/(|z_2|^2))`

By following the rules of exponent of a complex number, the root can be solved.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Exponent and Roots of Complex number**

• For `z_1^(z_2)`, convert `z_1` to polar form `re^(i theta)`

• For `a^(ib)`, convert `a` to `e^(ln a)` form

• For `z_1^(1/z_2)`, convert `1/z_2` to a complex number in numerator `bar(z_2)/(|z_2|^2)`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

What is `(1+i)^(3/2)`

- `3/2+3/2i`
- `2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))`
- `2^(3/2)(cos (3 pi/4)+i sin (3 pi/4))`
- `2^(3/4)(cos (3 pi/2)+i sin (3 pi/2))`

The answer is '`2^(3/4)(cos (3 pi/8)+i sin (3 pi/8))`'

*your progress details*

Progress

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Progress

Given z1 = a1+i b1 and z2 = a2+i b2, Can the exponent z1 to the power z2 be written in a+i b form?

no;exponent;power

No. Exponent to the power of complex number is not possible

yes;s;converting;by;buy

Yes. By converting to polar form z1 = r e power i theta.

The answer is 'Yes. By converting to polar form z1 = r e power i theta'.

Given z1 = a1+i b1 and z2 = a2+i b2, exponent ;; z1 power z2 ;; equals r1 e power i theta1 power a2+i b2 ;; equals r1 power a2 into r1 power i b2 into e power i theta1 a2 into e power i theta1 i b2 ;; equals r1 power a2 into e power i b2 ln r1 into e power i theta1 a2 into e power - theta1 b2 ;; equals r1 power a2 e power - theta1 b2 into e power i b2 ln r1 + theta1 a2 ;; The result is in the polar form and can be converted to coordinate form.

Given z1 = a1+i b1 and z2 = a2+i b2, Can the root z2 root of z1 be computed in a+i b form?

no;not;possible

No. root to a complex number is not possible

yes;s;considering;1

Yes. By considering root as power of one by z2

The answer is "Yes. By considering root as power of one by z2"

Given z1 = a1+i b1 and z2 = a2+i b2, ;; z2 root of z1 ;; equals z1 to the power 1 by z2 ;; equals z1 to the power conjugate of z2 by mod of z2 squared. ;; By following the rules of exponent of a complex number, the root can be solved.

To find exponent and root of complex numbers, the rules of numerical expression is used to arrive at the coordinate form a+i b

Exponent and Roots of Complex number: ;; For z1 power z2, convert z1 to polar form ;; For a power ib, convert A to e^ln a form ;; For z1 power 1 by z2, convert 1 by z2 to a complex number in numerator.

What is 1+ i power 3 by 2

by 2;buy 2

3 by 2 plus 3 by 2 i

pi by 8

2 power 3 by 4 cos 3 pi by 8 plus i sine 3 pi by 8

pi by 4

2 power 3 by 2 cos 3 pi by 4 plus i sine 3 pi by 4

pi by 2

2 power 3 by 4 cos 3 pi by 2 plus i sine 3 pi by 2

The answer is "2 power 3 by 4 cos 3 pi by 8 plus i sine 3 pi by 8"