If denotes the number of solutions of |1-i|^x=2^x and =(|z| (z)), where z=4(1+i)^4[1- i+i+-i1+ i], i=-1, then the distance of the point (, ) from the line 4 x-3 y=7 is __________.

aligned & (2)^x=2^x x=0 =1 \\ & z=4(1+i)^4 \\& [- i-i-+1+-i- i-1+] \\ & =- i2(1+4 i+6 i^2+4 i^3+1) \\ & =2 i \\ & =2 2=4 aligned Distance from (1,4)

If alpha denotes the number of solutions of 1ix2x and betafr: (Mathematics)

$\begin{aligned} & (\sqrt{2})^x=2^x \Rightarrow x=0 \Rightarrow \alpha=1 \\ & z=\frac{\pi}{4}(1+i)^4 \\& \left[\frac{\sqrt{\pi}-\pi i-i-\sqrt{\pi}}{\pi+1}+\frac{\sqrt{\pi}-i-\pi i-\sqrt{\pi}}{1+\pi}\right] \\ & =-\frac{\pi i}{2}\left(1+4 i+6 i^2+4 i^3+1\right) \\ & =2 \pi i \\ & \beta=\frac{2 \pi}{\frac{\pi}{2}}=4 \end{aligned}$

Distance from $(1,4)$ to $4 x-3 y=7 \\$ Will be $\frac{15}{5}=3$

Explanation

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