Question 1

Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$ ​
A) $x = 1,1 + i \sqrt { 3 } , 1 - i \sqrt { 3 }$
B) $x = 2 , - 2$
C) $x = - 2,2 - i \sqrt { 3 } , 2 + i \sqrt { 3 }$
D) $x = 2$
E) $x = 2 , - 1 + i \sqrt { 3 } , - 1 - i \sqrt { 3 }$

Accepted Answer

The equation $x^3 - 8 = 0$ can be solved by adding 8 to both sides and then taking the cube root of both sides, resulting in $x^3 = 8$. The real solution is $x = 2$, as $2^3 = 8$. However, this is a cubic equation, which means it has three roots in total, including complex roots. The other two roots can be found using the fact that the sum of the roots of a cubic equation with real coefficients must be real. The complex roots are the cube roots of unity multiplied by the real root, leading to $x = -1 + i\sqrt{3}$ and $x = -1 - i\sqrt{3}$, which are the two complex conjugate roots.

Question 2

Find all solutions to the following equation. $x ^ { 3 } + 729 = 0$ ​
A) $x = - 9$
B) $x = 9 i , - \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i , \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i$
C) $x = - 9 i$
D) $x = 9 i , - \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i , - \frac { 9 \sqrt { 3 } } { 2 } + \frac { 9 } { 2 } i$
E) $x = - 9 i , - \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i , - \frac { 9 \sqrt { 3 } } { 2 } + \frac { 9 } { 2 } i$

Accepted Answer

The answer of Find all solutions to the following equation....

Question 3

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$
A)​ $x = 6,6 + 6 i \sqrt { 3 } , 6 - 6 i \sqrt { 3 }$
B) $x = 6,3 - 3 i \sqrt { 3 } , - 3 - 3 i \sqrt { 3 }$
C)​ $x = 6 , - 3 - 3 i \sqrt { 3 } , - 3 + 3 i \sqrt { 3 }$ ​
D) $x = - 6 i$
E) $x = - 6$

Accepted Answer

The answer of Find all solutions to the following equation....

Question 4

Find the cube roots of the following complex number.Write each of the roots in standard form. -1
A) $- 5 , - \frac { 5 } { 2 } - \frac { 5 \sqrt { 3 } } { 2 } i , - \frac { 5 } { 2 } + \frac { 5 \sqrt { 3 } } { 2 } i$
B) $- 5,5$
C)-1
D) $- 5 , \frac { 5 } { 2 } - \frac { 5 \sqrt { 3 } } { 2 } i , \frac { 5 } { 2 } + \frac { 5 \sqrt { 3 } } { 2 } i$
E) $5 , - \frac { 5 } { 2 } + \frac { 5 \sqrt { 3 } } { 2 } i , - \frac { 5 } { 2 } - \frac { 5 \sqrt { 3 } } { 2 } i$

Accepted Answer

The answer of Find the cube roots of the following...

Quiz 40: Trigonometric Form of a Complex Number

Find all solutions to the following equation. x3−8=0x ^ { 3 } - 8 = 0x3−8=0 ​

Find all solutions to the following equation. x3+729=0x ^ { 3 } + 729 = 0x3+729=0 ​

Find all solutions to the following equation. x3−216=0x ^ { 3 } - 216 = 0x3−216=0

Find the cube roots of the following complex number.Write each of the roots in standard form. -1

Find all solutions to the following equation. $x ^ { 3 } - 8 = 0$

Find all solutions to the following equation. $x ^ { 3 } + 729 = 0$

Find all solutions to the following equation. $x ^ { 3 } - 216 = 0$