If-w-is-one-of-the-complex-cube-roots-of-unity-show-that-a-wb-w-2-c-a-w-2-b-wc-is-equal-to-2-b-2-c-2-ab-bc-c-Kindly-help-me-out-Thank-you-

Question Number 183667 by Mastermind last updated on 28/Dec/22

If w is one of the complex cube roots of unity, show that (a+wb+w^2 c)(a+w^2 b+wc) is equal to (α^2 +b^2 +c^2 −ab−bc−cα). Kindly help me out, Thank you.

$$\mathrm{If}\:\mathrm{w}\:\mathrm{is}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{cube}\: \\ $$$$\mathrm{roots}\:\mathrm{of}\:\mathrm{unity},\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{a}+\mathrm{wb}+\mathrm{w}^{\mathrm{2}} \mathrm{c}\right)\left(\mathrm{a}+\mathrm{w}^{\mathrm{2}} \mathrm{b}+\mathrm{wc}\right)\:\mathrm{is}\:\mathrm{equal} \\ $$$$\mathrm{to}\:\left(\alpha^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{ab}−\mathrm{bc}−\mathrm{c}\alpha\right). \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Kindly}\:\mathrm{help}\:\mathrm{me}\:\mathrm{out},\:\mathrm{Thank}\:\mathrm{you}. \\ $$

Commented by mr W last updated on 28/Dec/22

if you know (a+b)(c+d), then you can also solve this.

$${if}\:{you}\:{know}\:\left({a}+{b}\right)\left({c}+{d}\right),\:{then}\:{you}\:{can} \\ $$$${also}\:{solve}\:{this}. \\ $$

Commented by Frix last updated on 28/Dec/22

w^2 =conj w ⇒ w^2 +w=−1 w^3 =1 Should be easy now.

$${w}^{\mathrm{2}} =\mathrm{conj}\:{w}\:\Rightarrow\:{w}^{\mathrm{2}} +{w}=−\mathrm{1} \\ $$$${w}^{\mathrm{3}} =\mathrm{1} \\ $$$$\mathrm{Should}\:\mathrm{be}\:\mathrm{easy}\:\mathrm{now}. \\ $$

Commented by Rasheed.Sindhi last updated on 28/Dec/22

Typos in your question as usual. Corrected version: (a+ωb+ω^2 c)(a+ω^2 b+ωc) =a^2 +b^2 +c^2 −ab−bc−ca

$$\mathcal{T}{ypos}\:{in}\:{your}\:{question}\:{as}\:{usual}. \\ $$$${Corrected}\:{version}: \\ $$$$\left(\mathrm{a}+\omega\mathrm{b}+\omega^{\mathrm{2}} \mathrm{c}\right)\left(\mathrm{a}+\omega^{\mathrm{2}} \mathrm{b}+\omega\mathrm{c}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{ab}−\mathrm{bc}−\mathrm{ca} \\ $$

Commented by Mastermind last updated on 28/Dec/22

Thank you sir but i can′t see the solution

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}\: \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{can}'\mathrm{t}\:\mathrm{see}\:\mathrm{the}\:\mathrm{solution} \\ $$

Commented by Frix last updated on 28/Dec/22

Maybe the Master of the Mind is not the Master of the Formulas?

$$\mathrm{Maybe}\:\mathrm{the}\:\mathrm{Master}\:\mathrm{of}\:\mathrm{the}\:\mathrm{Mind}\:\mathrm{is}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{Master}\:\mathrm{of}\:\mathrm{the}\:\mathrm{Formulas}? \\ $$

Commented by Rasheed.Sindhi last updated on 10/Jan/23

✌️ Share your solution then

Commented by Mastermind last updated on 10/Jan/23

$$\mathrm{Smile}\: \\ $$$$\mathrm{Anyways},\:\mathrm{have}\:\mathrm{gotten}\:\mathrm{it}\:\mathrm{BOSS} \\ $$

Commented by Mastermind last updated on 19/Jan/23

Facebook Tweet Pin

If-w-is-one-of-the-complex-cube-roots-of-unity-show-that-a-wb-w-2-c-a-w-2-b-wc-is-equal-to-2-b-2-c-2-ab-bc-c-Kindly-help-me-out-Thank-you-

Leave a Reply Cancel reply