Question

how will you find the cube root of 6. Need details explanation​

Answer 1

Answer:

³√6=1.817

Step-by-step explanation:

z=x+iy

i=√-1

z³=(x+iy)³

=x³+3ix²y+3ixy²+i³y³

=x³+3ix²y-3xy²-iy³

=x(x²-3y²)+iy(3x²-y²)

z³=6

x(x²-3y²)=6

y(3x²-y²)=0

y=0

y²=3x²

-8x³=6

x=-frac12³√6~+/-0.90856

y=+/-√3x~+/-1.57367

-0.90856+1.57367i

i.e.,1.817(app.)

Answer 2

Step-by-step explanation:

6 does not have a cube root, it has three cube roots. Indeed, this is the case for every number, except for 0.

However, like all real numbers, 6 has only one real root; as someone else has already said is approximately 1.81712

What about the other roots? Well, we have to look at the complex plain for these.

Let us call one of these roots z=x+iyz=x+iy , where i=−1−−−√i=−1

If zz is a cube root of 6, then z3=(x+iy)3z3=(x+iy)3

=x3+3ix2y+3i2xy2+i3y3=x3+3ix2y+3i2xy2+i3y3

=x3+3ix2y−3xy2−iy3=x3+3ix2y−3xy2−iy3

=x(x2−3y2)+iy(3x2−y2)=x(x2−3y2)+iy(3x2−y2)

As z3=6z3=6 ,

x(x2−3y2)=6x(x2−3y2)=6

y(3x2−y2)=0→y=0y(3x2−y2)=0→y=0 OR y2=3x2y2=3x2

If y=0y=0 , then zz is the real number xx ; we already know this is 1.81712

If y2=3x2y2=3x2 , then (from 1.), −8x3=6→x=−frac126–√3≈−0.90856−8x3=6→x=−frac1263≈−0.90856

From 2., y=±3–√x≈±1.57367y=±3x≈±1.57367

The cube roots (to 6 decimal plsces) are thus:

1.817121.81712

−0.90856+1.57367i−0.90856+1.57367i

−0.90856−1.57367i−0.90856−1.57367i