a) Find the square roots of $3+4i$.
b) Find all cube roots of 8i.
where i=sqrt-1
Algrebra Help
a) Find the square roots of 3+4i.
z=a+bi=3+4i|z|=√a2+b2=√32+42=5sin(φ)=b|z|=45=0.8φ=arcsin(0.8)+2kπz=|z|eiφz=5ei⋅(arcsin(0.8)+2kπ)
√3+4i=(3+4i)12=512(ei⋅(arcsin(0.8)+2kπ))12=√5⋅ei⋅(arcsin(0.8)+2kπ2)=√5⋅ei⋅(arcsin(0.8)2+kπ)|k=(0,1)√3+4i=√5⋅ei⋅(arcsin(0.8)2)|k=0=√5⋅(cos(arcsin(0.8)2)+i⋅sin(arcsin(0.8)2))=2.23606797750⋅(0.89442719100+i⋅0.44721359550)=2+i⋅1√3+4i=√5⋅ei⋅(arcsin(0.8)2+π)|k=1=√5⋅(cos(arcsin(0.8)2+π)+i⋅sin(arcsin(0.8)2+π))=√5⋅(−cos(arcsin(0.8)2)−i⋅sin(arcsin(0.8)2))=2.23606797750⋅(−0.89442719100−i⋅0.44721359550)=−2−i⋅1
All 2nd roots of 3+4i :
2+i−2−i
Algrebra Help
b) Find all cube roots of 8i.
3√8i=3√8⋅3√i=3√23⋅3√i=2⋅3√i
z=a+bi=i|z|=√02+12=√1=1sin(φ)=b|z|=11=1φ=arcsin(1)+2kπφ=π2+2kπz=|z|eiφ=1⋅ei⋅(π2+2kπ)z=ei⋅(π2+2kπ)
3√i=(i)13=(ei⋅(π2+2kπ))13=ei⋅(π2+2kπ3)=ei⋅(π6+23⋅kπ)|k=(0,1,2)3√i=ei⋅(π6)|k=0=cos(π6)+i⋅sin(π6)|π6=30∘=cos(30∘)+i⋅sin(30∘)=√32+i⋅122⋅3√i=2⋅(√32+i⋅12)2⋅3√i=√3+i3√i=ei⋅(π6+23π)|k=1=ei⋅(5π6)=cos(5π6)+i⋅sin(5π6)|5π6=150∘=cos(150∘)+i⋅sin(150∘)=−cos(30∘)+i⋅sin(30∘)=−√32+i⋅122⋅3√i=2⋅(−√32+i⋅12)2⋅3√i=−√3+i3√i=ei⋅(π6+2⋅23π)|k=2=ei⋅(3π2)=cos(3π2)+i⋅sin(3π2)|3π2=270∘=cos(270∘)+i⋅sin(270∘)=0+i⋅sin(90∘)=−i2⋅3√i=2⋅(−i)2⋅3√i=−2i
All 3rd roots of 8i :
√3+i−√3+i−2i
Simplify the following:
sqrt(4 i + 3)
Express 4 i + 3 as a square using 4 i + 3 = 4 + 4 i + i^2, then look to factor.
3 + 4 i = 4 + 4 i - 1 = 4 + 4 i + i^2 = (i + 2)^2:
sqrt((i + 2)^2 )
For all complex z with Re(z)>0, sqrt(z^2) = z.
Cancel exponents. sqrt((2 + i)^2) = i + 2:
2 + i and -2 - i
Simplify the following:
(8 i)^(1/3)
Express 8 i as a cube using 8 i = 3 sqrt(3) + 9 i + 3 sqrt(3) i^2 + i^3, then look to factor.
8 i = 3 sqrt(3) + 9 i - 3 sqrt(3) - i = 3 sqrt(3) + 9 i + 3 sqrt(3) i^2 + i^3 = (sqrt(3))^3 + 3 (sqrt(3))^2 i + 3 sqrt(3) i^2 + i^3 = (sqrt(3) + i)^3:
((sqrt(3) + i)^3 )^(1/3)
For all complex z with -π/3 Cancel exponents. ((sqrt(3) + i)^3)^(1/3) = sqrt(3) + i:
sqrt(3) + i -sqrt(3) + i -2i