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find the polar of (1+i)(1+i(√3))

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{polar}}\:\boldsymbol{\mathrm{of}} \\ $$$$\left(\mathrm{1}+\boldsymbol{\mathrm{i}}\right)\left(\mathrm{1}+\boldsymbol{\mathrm{i}}\sqrt{\mathrm{3}}\right) \\ $$

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Question Number 226562 Answers: 1 Comments: 0

If x + y i = ((a + i)/(a − i)) , prove that ay − 1 = x. (x+yi)(a−i)=a+i ax − xi + ayi − yi^2 = a + i (ax + y) + (ay − x)i = a + i ay − x = 1 x = ay −1

$$\mathrm{If}\:\mathrm{x}\:+\:\mathrm{y}\:{i}\:=\:\frac{\mathrm{a}\:+\:{i}}{\mathrm{a}\:−\:{i}}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{ay}\:−\:\mathrm{1}\:=\:\mathrm{x}. \\ $$$$\:\left(\mathrm{x}+\mathrm{y}{i}\right)\left(\mathrm{a}−{i}\right)=\mathrm{a}+{i} \\ $$$$\:\:\:\mathrm{ax}\:−\:\mathrm{x}{i}\:+\:\mathrm{ay}{i}\:−\:\mathrm{y}{i}^{\mathrm{2}} \:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\left(\mathrm{ax}\:+\:\mathrm{y}\right)\:+\:\left(\mathrm{ay}\:−\:\mathrm{x}\right){i}\:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\mathrm{ay}\:−\:\mathrm{x}\:=\:\mathrm{1} \\ $$$$\:\:\mathrm{x}\:=\:\mathrm{ay}\:−\mathrm{1} \\ $$

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a+b+c = x lim_(x→0) ((a^3 +b^3 +c^3 )/(abc)) =?

$$\:\:\:\: {a}+{b}+{c}\:=\:{x}\: \\ $$$$\:\:\:\: \underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }{{abc}}\:=? \\ $$

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Parametric Surface r^ (u,v);R^2 →R^3 r^ (u,v)= { ((a∙sin(u)cos(v))),((a∙sin(u)sin(v))),((a∙cos(u))) :} a>0 , 0≤u≤π , 0≤v≤2π 1. Find Principal Direction 2. Find Principal Curvature 3. Find Gauss Curvature 4. Find Euler Characteristic Hint Shape Operator S=r_(,μν) ^λ ∗N_λ ^

$$\mathrm{Parametric}\:\mathrm{Surface}\:\hat {\boldsymbol{\mathrm{r}}}\left({u},{v}\right);\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\hat {\boldsymbol{\mathrm{r}}}\left({u},{v}\right)=\begin{cases}{{a}\centerdot\mathrm{sin}\left({u}\right)\mathrm{cos}\left({v}\right)}\\{{a}\centerdot\mathrm{sin}\left({u}\right)\mathrm{sin}\left({v}\right)}\\{{a}\centerdot\mathrm{cos}\left({u}\right)}\end{cases}\:\:\:{a}>\mathrm{0}\:,\:\mathrm{0}\leq{u}\leq\pi\:,\:\mathrm{0}\leq{v}\leq\mathrm{2}\pi \\ $$$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Direction} \\ $$$$\mathrm{2}.\:\mathrm{Find}\:\mathrm{Principal}\:\mathrm{Curvature} \\ $$$$\mathrm{3}.\:\mathrm{Find}\:\mathrm{Gauss}\:\mathrm{Curvature} \\ $$$$\mathrm{4}.\:\mathrm{Find}\:\mathrm{Euler}\:\mathrm{Characteristic}\: \\ $$$$\mathrm{Hint}\: \\ $$$$\mathrm{Shape}\:\mathrm{Operator}\:\mathcal{S}={r}_{,\mu\nu} ^{\lambda} \ast\hat {\mathcal{N}}_{\lambda} \: \\ $$

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Question Number 226569 Answers: 1 Comments: 2

a^4 + b^4 + c^4 = 2d^2 Prove that the equation has an infinite number of natural solutions

$$\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:=\:\mathrm{2d}^{\mathrm{2}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{solutions} \\ $$

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If (x+(2a^2 +5))(x−(2a^2 +7)) ≤ 0 x∈[−(a^2 +8a−10) ; (a^2 +9a−11)] Find: a = ?

$$\mathrm{If}\:\:\:\left(\mathrm{x}+\left(\mathrm{2a}^{\mathrm{2}} +\mathrm{5}\right)\right)\left(\mathrm{x}−\left(\mathrm{2a}^{\mathrm{2}} +\mathrm{7}\right)\right)\:\leqslant\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{x}\in\left[−\left(\mathrm{a}^{\mathrm{2}} +\mathrm{8a}−\mathrm{10}\right)\:;\:\left(\mathrm{a}^{\mathrm{2}} +\mathrm{9a}−\mathrm{11}\right)\right] \\ $$$$\mathrm{Find}:\:\boldsymbol{\mathrm{a}}\:=\:? \\ $$

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