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heureka

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Questions 17
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 #3
avatar+26396 
+10

(1+i)7?

z=x+iy  z=1+ix=1,y=+1

r=x2+y2=(1)2+12=2

ϕ=sign(y)cos1(xr)=+cos1(12)=cos1(22)=34π

z=reiϕ

\\z=\sqrt2*e^{i*\frac{3}{4}\pi} \\  z^7=\left(\sqrt2*e^{i*\frac{3}{4}\pi}\right)^7 \\  z^7=\left(\sqrt2\right)^7*\left(e^{i*\frac{3}{4}\pi}\right)^7 \\  z^7=\left(\sqrt2\right)^7*e^{i*\frac{7*3}{4}\pi} \\  z^7=\left(\sqrt2\right)^7*e^{i*\frac{21}{4}\pi}   = \left(\sqrt2\right)^7*e^{i\left(4\pi+\pi+\frac{1}{4}\pi\textcolor[rgb]{1,0,0}{-4\pi}}\right)\\

z^7=\left(\sqrt2\right)^7*e^{i\left(\pi+\frac{1}{4}\pi}\right)\\

eiϕ=cosϕ+isinϕ

z7=(2)7[cos(π+π4)+isin(π+π4)]z7=(2)7[cosπ1cosπ4sinπ0sinπ4+i(sinπ0cosπ4+cosπ1sinπ4)]z7=(2)7(cosπ4isinπ4)

\sin{\frac{\pi}{4}}=\cos{\frac{\pi}{4}}=\frac{\sqrt{2} }{2}\quad\\  z^7=\left(\sqrt2\right)^7 \left( -\frac{\sqrt{2}}{2}-i{\frac{\sqrt{2}}{2}\right)\\  z^7=\left(\sqrt2\right)^7\frac{\sqrt{2}}{2}\left( -1-i\right)\\  z^7=\frac{\left(\sqrt{2}\right)^8}{2}\left(-1-i\right)

z7=242(1i)z7=23(1i)z7=8(1i)z7=88i

(1+i)7=88i

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28 mai 2014
 #3
avatar+26396 
+10

 

\boxed{2^{\frac{x}{2}} * 4^{\frac{x}{6}}* \left[(\frac{1}{8})^{\frac{1}{x}}\right]^{\frac{1}{6}}=2^2*2^{\frac{1}{3}} }\\\\  \Rightarrow 2^{\frac{x}{2}} *2^{2*\frac{x}{6}}*2^{-3*\frac{1}{x}*\frac{1}{6}}=2^{2+\frac{1}{3}}\\\\   \Rightarrow 2^{\frac{x}{2}} *2^{\frac{x}{3}}*2^{\left(-\frac{1}{2x}\right)}=2^{\frac{7}{3}}\\\\  \Rightarrow 2^{  \left(\frac{x}{2}+\frac{x}{3}-\frac{1}{2x}\right)  }=2^{\frac{7}{3}} \quad | \quad ln\\\\   \Rightarrow {  \left(\frac{x}{2}+\frac{x}{3}-\frac{1}{2x}\right)*\ln(2)  }=\frac{7}{3}*\ln(2)\\\\   \Rightarrow {  \frac{x}{2}+\frac{x}{3}-\frac{1}{2x}\right  }=\frac{7}{3}\\\\

\\\Rightarrow {  \frac{3x}{3x}*\frac{x}{2}+\frac{2x}{2x}*\frac{x}{3}  -\frac{3}{3}*\frac{1}{2x}\right   }=\frac{7}{3}\\\\   \Rightarrow \frac{3x^2+2x^2-3}{6x}=\frac{7}{3} \quad | \quad *6x\\\\   \Rightarrow 5x^2-3=6x*\frac{7}{3}\\  \Rightarrow 5x^2-3=14x\\

5x214x3=0x2145x35=0x1,2=1425±1414(25)(25)+35x1,2=1410±1414(10)(10)+202035x1,2=1410±196100+60100x1,2=1410±196+60100x1,2=1410±256100x1,2=1410±1610x1=1410+1610=3010=3x2=14101610=210=15

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28 mai 2014