To solve the equation X^6 + 8 = 0, we can follow these steps:
Step 1: Subtract 8 from both sides of the equation to isolate the term with the variable:
X^6 = -8
Step 2: Take the sixth root of both sides to solve for X. The sixth root of a number is the number that, when raised to the power of 6, gives the original number:
X = (-8)^(1/6)
Step 3: Simplify the expression on the right side of the equation. Since (-8)^(1/6) is a complex number, it can be expressed in multiple ways. One common way is to write it in polar form:
X = 2 * cos((π + 2kπ)/6) + i * sin((π + 2kπ)/6)
where k is an integer ranging from 0 to 5.
Step 4: Evaluate the expression for each value of k to obtain all possible solutions. Substituting k = 0, 1, 2, 3, 4, 5 into the equation will give us the different values of X.
For example, when k = 0:
X = 2 * cos(π/6) + i * sin(π/6)
X = 2 * (√3/2) + i * (1/2)
X = √3 + i/2
Similarly, when k = 1:
X = 2 * cos(π/3) + i * sin(π/3)
X = 2 * (1/2) + i * (√3/2)
X = 1 + √3i/2
And so on, for k = 2, 3, 4, and 5.
These are the different solutions for the equation X^6 + 8 = 0. Each value of X corresponds to a complex number on the complex plane.To solve the equation X^6 + 8 = 0, we can follow these steps:
Step 1: Subtract 8 from both sides of the equation to isolate the term with the variable:
X^6 = -8
Step 2: Take the sixth root of both sides to solve for X. The sixth root of a number is the number that, when raised to the power of 6, gives the original number:
X = (-8)^(1/6)
Step 3: Simplify the expression on the right side of the equation. Since (-8)^(1/6) is a complex number, it can be expressed in multiple ways. One common way is to write it in polar form:
X = 2 * cos((π + 2kπ)/6) + i * sin((π + 2kπ)/6)
where k is an integer ranging from 0 to 5.
Step 4: Evaluate the expression for each value of k to obtain all possible solutions. Substituting k = 0, 1, 2, 3, 4, 5 into the equation will give us the different values of X.
For example, when k = 0:
X = 2 * cos(π/6) + i * sin(π/6)
X = 2 * (√3/2) + i * (1/2)
X = √3 + i/2
Similarly, when k = 1:
X = 2 * cos(π/3) + i * sin(π/3)
X = 2 * (1/2) + i * (√3/2)
X = 1 + √3i/2
And so on, for k = 2, 3, 4, and 5.
These are the different solutions for the equation X^6 + 8 = 0. Each value of X corresponds to a complex number on the complex plane.