Simplify by factoring

Please show work! I need to learn how to do these type of problems by looking at the work.

Simplify: (x^(2/3) + x^(-1/3)) / (x^2-1)

Simplify: (1/x + 3) / (1/5 - 1/x^2)

Simplify: (x^3 + 8) / (x^3 + 5x^2 + 14x)

Simplify: {(x-3) / (x-4)} + {(2x) / (x^2-6x+8)} -5

Find all roots (real & imaginary): x^3 - 27 = 0

Thanks! Best answer to whoever shows work and answers all questions!

1. Factor the top expression by x^(-1/3) to get:

x^(-1/3)(x^(2/3 + 1/3) + x^(-1/3 + 1/3))/(x² - 1)
= x^(-1/3)(x^(3/3) + x^(0))/(x² - 1)
= x^(-1/3)(x + 1)/(x² - 1)

Then, factor the bottom expression and reduce the common factors of x + 1 to get:

x^(-1/3)(x + 1)/((x + 1)(x - 1))
= x^(-1/3)/(x - 1)

We can rewrite the expression as:

1/(x^(1/3)(x - 1))

2. Multiply the top and bottom expression by 5x² to get:

(5x + 15x²)/(x² - 5)

We can factor the top expression by the GCF of 5x to get:

5x(1 + 3x)/(x² - 5)

3. Factor each expression!

For x³ + 8, use sum of cubes, which is a³ + b³ = (a + b)(a² - ab + b²), so we have (x + 2)(x² - 2x + 4).
For x³ + 5x² + 14x, factor the GCF of x to get x(x² + 5x + 14). Therefore, we obtain:

(x + 2)(x² - 2x + 4)/(x(x² + 5x + 14))

4. (x - 3)/(x - 4) + 2x/(x² - 6x + 8) - 5

Factor x² - 6x + 8:

(x - 3)/(x - 4) + 2x/((x - 4)(x - 2)) - 5

Then, since LCD is (x - 4)(x - 2), multiply the top and bottom of each term by its missing common denominator.

(x - 3)(x - 2)/((x - 4)(x - 2)) + 2x/((x - 4)(x - 2)) - 5(x - 4)(x - 2)/((x - 4)(x - 2))

Finally, combine those expressions:

(x² - 5x + 6 + 2x - 5(x² - 6x + 8))/((x - 4)(x - 2))
= (x² - 3x + 6 - 5x² + 30x - 40)/((x - 4)(x - 2))
= (-4x² + 27x - 36)/((x - 4)(x - 2))

5. Add both sides by 27:

x³ - 27 + 27 = 27
x³ = 27

Finally, set both sides to the power of 1/3 to get:

(x³)^(1/3) = (27)^(1/3)
x = (3³)^(1/3)
x = 3

I hope this helps!