Solution:

To find the value of p^3 * q^3, we need to first find the values of p and q.

The given equation is X^2 + 2X + 1 = 0.

We can solve this quadratic equation by factoring or by using the quadratic formula.

Factoring:

The equation can be factored as (X + 1)(X + 1) = 0.

So, the roots of the equation are X = -1 and X = -1.

Quadratic formula:

Using the quadratic formula, X = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

In this case, a = 1, b = 2, and c = 1.

Using the quadratic formula, we get X = (-2 ± √(2^2 - 4*1*1)) / 2*1.

Simplifying this, we get X = (-2 ± √(4 - 4)) / 2.

X = (-2 ± √0) / 2.

Since the discriminant is zero, the roots are equal.

So, X = -1 and X = -1.

Calculating p^3 * q^3:

Now that we know the values of p and q, which are -1 and -1, respectively, we can calculate p^3 * q^3.

p^3 * q^3 = (-1)^3 * (-1)^3.

Simplifying this, we get -1 * -1 * -1 * -1.

Multiplying these values, we get 1.

Therefore, the value of p^3 * q^3 is 1.

Summary:

The roots of the given quadratic equation X^2 + 2X + 1 = 0 are -1 and -1. Therefore, the value of p^3 * q^3 is 1.