If-x-yi-4-a-bi-show-that-a-2-b-2-x-2-y-2-4-

Question Number 168208 by MathsFan last updated on 06/Apr/22

I f {(x + y i)}^{4} = a + b i,

s h o w t h a t a^{2} + b^{2} = {(x^{2} + y^{2})}^{4}

Commented by MJS_new last updated on 06/Apr/22

you can do it yourself

(1) expand {(x + y i)}^{4} = term 1

(2) a = real (term 1) \land b = imag (term 1)

(3) expand a^{2} + b^{2} = term 2

(4) sort term 2 and you should see the

factors 1 4 6 4 1 \Rightarrow there you are

Answered by mr W last updated on 06/Apr/22

g e n e r a l l y w e h a v e

i f z_{1} = z_{2}, t h e n ∣ z_{1} ∣=∣ z_{2} ∣ .

i f z_{1} = z_{2}^{n}, t h e n ∣ z_{1} ∣=∣ z_{2} ∣^{n} .

a p p l y i n g t h i s w e g e t

f r o m {(x + y i)}^{4} = a + b i

∣ x + y i ∣^{4} =∣ a + b i ∣, o r

{(\sqrt{x^{2} + y^{2}})}^{4} = \sqrt{a^{2} + b^{2}}, o r

{(x^{2} + y^{2})}^{4} = a^{2} + b^{2}

Commented by MathsFan last updated on 19/Apr/22

t h a n k y o u s i r

Answered by Mathspace last updated on 06/Apr/22

{(x + y i)}^{4} = a + b i \Rightarrow {(x - y i)}^{4} = a - b i \Rightarrow

(a + b i) (a - b i) = {(x + y i)}^{4} {(x - y i)}^{4} \Rightarrow

a^{2} + b^{2} = {((x + y i) (x - y i))}^{4} \Rightarrow

a^{2} + b^{2} = {(x^{2} + y^{2})}^{4}

Commented by mr W last updated on 07/Apr/22

h o w d o y o u g e t

{(x + y i)}^{4} = a + b i \Rightarrow {(x - y i)}^{4} = a - b i ?

i t i s n o t o b v i o u s .

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