Please explain the steps on to solve this equation.

x^-6 - 8x^-3 + 16 = 0

k^-4/3 - 10k^-2/3 + 21 = 0

x^-6 - 8x^-3 + 16 = 0

Put x^(-3) = y. Then:

y^2 - 8 y + 16 = 0 --->

(y - 4)^2 = 0 --->

y = 4 ---->

x = 4^(-1/3) = 2^(-2/3)

To solve the first equation, x^-6 - 8x^-3 + 16 = 0, we can follow these steps:

Step 1: Start by defining a substitution. Let's say u = x^-3. This substitution will simplify the equation.

Step 2: Substitute the value of u in the equation. We have u^2 - 8u + 16 = 0.

Step 3: Now we can solve this quadratic equation. We can use factoring or the quadratic formula to find the solutions. In this case, the equation can be factored as (u - 4)^2 = 0.

Step 4: Solve for u by taking the square root of both sides of the equation. We get u - 4 = 0, which simplifies to u = 4.

Step 5: Substitute back the value of u in terms of x. Since u = x^-3, we have x^-3 = 4.

Step 6: Take the reciprocal of both sides to solve for x. This gives us x^3 = 1/4.

Step 7: Finally, take the cube root of both sides to find the value of x. We have x = (1/4)^(1/3), which simplifies to x = (1/2)^(1/3) or x = 0.7937 (rounded to four decimal places).

To solve the second equation, k^-4/3 - 10k^-2/3 + 21 = 0, we can follow a similar process:

Step 1: Define a substitution. Let's say u = k^-2/3.

Step 2: Substitute the value of u in the equation. We have u^2 - 10u + 21 = 0.

Step 3: Solve this quadratic equation. Factor the quadratic equation (u - 3)(u - 7) = 0.

Step 4: Solve for u by setting each factor equal to zero. We have u - 3 = 0 or u - 7 = 0, which simplifies to u = 3 or u = 7.

Step 5: Substitute back the value of u in terms of k. Since u = k^-2/3, we have k^-2/3 = 3 or k^-2/3 = 7.

Step 6: Take the reciprocal of both sides to solve for k. This gives us k^2/3 = 1/3 or k^2/3 = 1/7.

Step 7: Cube both sides of the equation to find the value of k. We have k = (1/3)^(3/2) or k = (1/7)^(3/2), which simplifies to k = 0.5774 or k = 0.1826 (rounded to four decimal places).