Question

Which shows all of the solutions of the equation y3

?

216

O y= }

1

3

O y=+;

1

O y = 108

o y= +108

Answer 1

The equation y^3 = 216 can be solved by finding the cube root of both sides. Since there are three complex cube roots of 216, the solutions of the equation can be expressed in the polar form as:

y = 6(cos(2πk/3) + i sin(2πk/3)), where k = 0, 1, or 2

Using the cosine and sine values for angles 0, 120, and 240 degrees (or 0, 2π/3, and 4π/3 radians), we can write the solutions in rectangular form as follows:

y = 6(cos(0) + i sin(0)) = 6(1 + 0i) = 6

y = 6(cos(2π/3) + i sin(2π/3)) = 6(-1/2 + i √3/2) = -3 + 3√3 i

y = 6(cos(4π/3) + i sin(4π/3)) = 6(-1/2 - i √3/2) = -3 - 3√3 i

Therefore, the solutions of the equation y^3 = 216 are:

y = 6, -3 + 3√3 i, and -3 - 3√3 i

None of the given options show all of these solutions.

Explanation:

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