For the numbers in
form, convert to polar form:
![1+i=\sqrt2\dfrac{1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)](https://tex.z-dn.net/?f=1%2Bi%3D%5Csqrt2%5Cdfrac%7B1%2Bi%7D%7B%5Csqrt2%7D%3D%5Csqrt2%5Cleft%28%5Ccos%5Cdfrac%5Cpi4%2Bi%5Csin%5Cdfrac%5Cpi4%5Cright%29)
By DeMoivre's theorem,
![(1+i)^5=(\sqrt2)^5\left(\cos\dfrac{5\pi}4+i\sin\dfrac{5\pi}4\right)=4\sqrt2\dfrac{-1-i}{\sqrt2}=-4-4i](https://tex.z-dn.net/?f=%281%2Bi%29%5E5%3D%28%5Csqrt2%29%5E5%5Cleft%28%5Ccos%5Cdfrac%7B5%5Cpi%7D4%2Bi%5Csin%5Cdfrac%7B5%5Cpi%7D4%5Cright%29%3D4%5Csqrt2%5Cdfrac%7B-1-i%7D%7B%5Csqrt2%7D%3D-4-4i)
![-1+i=\sqrt2\dfrac{-1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4}\right)](https://tex.z-dn.net/?f=-1%2Bi%3D%5Csqrt2%5Cdfrac%7B-1%2Bi%7D%7B%5Csqrt2%7D%3D%5Csqrt2%5Cleft%28%5Ccos%5Cdfrac%7B3%5Cpi%7D4%2Bi%5Csin%5Cdfrac%7B3%5Cpi%7D4%7D%5Cright%29)
![\implies(-1+i)^6=(\sqrt2)^6\left(\cos\dfrac{18\pi}4+i\sin\dfrac{18\pi}4\right)=8i](https://tex.z-dn.net/?f=%5Cimplies%28-1%2Bi%29%5E6%3D%28%5Csqrt2%29%5E6%5Cleft%28%5Ccos%5Cdfrac%7B18%5Cpi%7D4%2Bi%5Csin%5Cdfrac%7B18%5Cpi%7D4%5Cright%29%3D8i)
![\sqrt3+i=2\dfrac{\sqrt3+i}2=2\left(\cos\dfrac\pi6+i\sin\dfrac\pi6\right)](https://tex.z-dn.net/?f=%5Csqrt3%2Bi%3D2%5Cdfrac%7B%5Csqrt3%2Bi%7D2%3D2%5Cleft%28%5Ccos%5Cdfrac%5Cpi6%2Bi%5Csin%5Cdfrac%5Cpi6%5Cright%29)
![\implies2(\sqrt3+i)^{10}=2^{11}\left(\cos\dfrac{10\pi}6+i\sin\dfrac{10\pi}6\right)=2^{11}\dfrac{1-i\sqrt3}2=2^{10}(1-i\sqrt3)](https://tex.z-dn.net/?f=%5Cimplies2%28%5Csqrt3%2Bi%29%5E%7B10%7D%3D2%5E%7B11%7D%5Cleft%28%5Ccos%5Cdfrac%7B10%5Cpi%7D6%2Bi%5Csin%5Cdfrac%7B10%5Cpi%7D6%5Cright%29%3D2%5E%7B11%7D%5Cdfrac%7B1-i%5Csqrt3%7D2%3D2%5E%7B10%7D%281-i%5Csqrt3%29)
For the numbers already in polar form, DeMoivre's theorem can be applied directly:
![2\left(\cos20^\circ+i\sin20^\circ\right)^3=2\left(\cos60^\circ+i\sin60^\circ\right)=2\dfrac{1+i\sqrt3}2=1+i\sqrt3](https://tex.z-dn.net/?f=2%5Cleft%28%5Ccos20%5E%5Ccirc%2Bi%5Csin20%5E%5Ccirc%5Cright%29%5E3%3D2%5Cleft%28%5Ccos60%5E%5Ccirc%2Bi%5Csin60%5E%5Ccirc%5Cright%29%3D2%5Cdfrac%7B1%2Bi%5Csqrt3%7D2%3D1%2Bi%5Csqrt3)
![2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)^4=2(\cos\pi+i\sin\pi)=-2](https://tex.z-dn.net/?f=2%5Cleft%28%5Ccos%5Cdfrac%5Cpi4%2Bi%5Csin%5Cdfrac%5Cpi4%5Cright%29%5E4%3D2%28%5Ccos%5Cpi%2Bi%5Csin%5Cpi%29%3D-2)
At second glance, I think the 2s in the last two numbers should also be getting raised to the 3rd and 4th powers:
![\left(2(\cos20^\circ+i\sin20^\circ)\right)^3=8\left(\cos60^\circ+i\sin60^\circ\right)=4+4\sqrt3](https://tex.z-dn.net/?f=%5Cleft%282%28%5Ccos20%5E%5Ccirc%2Bi%5Csin20%5E%5Ccirc%29%5Cright%29%5E3%3D8%5Cleft%28%5Ccos60%5E%5Ccirc%2Bi%5Csin60%5E%5Ccirc%5Cright%29%3D4%2B4%5Csqrt3)
![\left(2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)\right)^4=16(\cos\pi+i\sin\pi)=-16](https://tex.z-dn.net/?f=%5Cleft%282%5Cleft%28%5Ccos%5Cdfrac%5Cpi4%2Bi%5Csin%5Cdfrac%5Cpi4%5Cright%29%5Cright%29%5E4%3D16%28%5Ccos%5Cpi%2Bi%5Csin%5Cpi%29%3D-16)
Answer:
x<-5
Step-by-step explanation:
-0.4x>0.8+1.2
-0.4x>2
x<-5
Answer:
(-6,1)
Step-by-step explanation:
The rule for reflecting over the X axis is to negate the value of the y-coordinate of each point, but leave the x-value the same.
Answer:
x=-2/15
Step-by-step explanation:
Answer:
Luis spent $10 in total
2×2=4 6×1=6 6+4=10
Step-by-step explanation: