+26 Find $\left(\frac{1+i}{\sqrt{2}}\right)^{46}$. Any help is appreciated! Thanks!
+816 In \(\left(\frac{1+i}{\sqrt{2}}\right)^{46}\) , we can apply exponent rule, \((\frac{a}{b})^c=\frac{a^c}{b^c}\) . If we do it here, we will get: \(\frac{(1+i)^46}{2^{23}}\) .\(\)
+9465 Find \(\left(\frac{1+i}{\sqrt{2}}\right)^{46}\) .
Let's convert 1 + i to polar form so we can use de Moivre's Theorem.
\(1+i=\sqrt2(\cos\frac{\pi}4+i\sin\frac{\pi}4)\)
Now lets raise both sides of that equation to the power of 46 .
\((1+i)^{46}=[\sqrt2(\cos\frac{\pi}4+i\sin\frac{\pi}4)]^{46}\\~\\ (1+i)^{46}=(\sqrt2\,)^{46}(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}\)
And divide both sides by \((\sqrt2\,)^{46}\)
\(\frac{(1+i)^{46}}{(\sqrt2\,)^{46}}=(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}\\~\\ \left(\frac{1+i}{\sqrt{2}}\right)^{46}=(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}\)
De Moivre's Theorem says....
\((\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)\) So....
\((\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}=\cos(\frac{46\pi}{4})+i\sin(\frac{46\pi}{4})\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=\cos(\frac{46\pi}{4})+i\sin(\frac{46\pi}{4})\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=\cos(\frac{23\pi}{2})+i\sin(\frac{23\pi}{2}) \)
Now let's find a reference angle that is coterminal with \(\frac{23\pi}{2}\) by subtracting \(2\pi\) five times.
\(\frac{23\pi}{2}-2\pi-2\pi-2\pi-2\pi-2\pi\,=\,\frac{23\pi}{2}-10\pi\,=\,\frac{23\pi}{2}-\frac{20\pi}{2}\,=\,\frac{3\pi}{2}\)
So....
\(\left(\frac{1+i}{\sqrt2}\right)^{46}=\cos(\frac{3\pi}{2})+i\sin(\frac{3\pi}{2})\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=0+i(-1)\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=-i \)
+26367 Find
\(\large{\left(\dfrac{1+i}{\sqrt{2}}\right)^{46}}\)
\left(\frac{1+i}{\sqrt{2}}\right)^{46}.
\(\begin{array}{|rcll|} \hline && \left(\dfrac{1+i}{\sqrt{2}}\right)^{46} \\\\ &=& \dfrac{ \left(1+i \right)^{46} }{ \left(\sqrt{2} \right)^{46} } \\\\ &=& \dfrac{ \left( \left(1+i \right)^{2}\right)^{23} }{ 2^{\frac{46}{2}} } \quad & | \quad \left(1+i \right)^{2} = 2i \\\\ &=& \dfrac{ \left( 2i \right)^{23} } { 2^{23} } \\\\ &=& \dfrac{ 2^{23}i^{23} } { 2^{23} } \\\\ &=& i^{23} \\\\ &=& \left(i^{2}\right)^{11}i \quad & | \quad i^2 = -1 \\\\ &=& \left(-1 \right)^{11}i \\\\ &\mathbf{=}& \mathbf{-i} \\ \hline \end{array}\)
