To solve the equation (x - 3)^7 - (x - 5)^5 = 0, we can use the binomial theorem. The binomial theorem states that for any positive integer n:

(a + b)^n = C(n, 0)a^n + C(n, 1)a^(n-1)b + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)ab^(n-1) + C(n, n)b^n

where C(n, k) represents the binomial coefficient, which is the number of ways to choose k items from a set of n items.

In this case, we have (x - 3)^7 - (x - 5)^5 = 0. Expanding using the binomial theorem, we get:

C(7, 0)(x^7) + C(7, 1)(x^6)(-3) + C(7, 2)(x^5)(-3)^2 + C(7, 3)(x^4)(-3)^3 + C(7, 4)(x^3)(-3)^4 + C(7, 5)(x^2)(-3)^5 + C(7, 6)(x)(-3)^6 + C(7, 7)(-3)^7 - C(5, 0)(x^5) + C(5, 1)(x^4)(-5) + C(5, 2)(x^3)(-5)^2 + C(5, 3)(x^2)(-5)^3 + C(5, 4)(x)(-5)^4 - C(5, 5)(-5)^5 = 0

Simplifying this equation, we get:

x^7 - 21x^6 + 189x^5 - 903x^4 + 2380x^3 - 3360x^2 + 2520x - 729 - x^5 + 5x^4 - 25x^3 + 125x^2 - 625x + 3125 = 0

Combining like terms, we have:

x^7 - 21x^6 + 188x^5 - 928x^4 + 2455x^3 - 3235x^2 + 1895x + 2396 = 0

Unfortunately, this equation does not have a simple solution. It may be necessary to use numerical methods or approximation techniques to find an approximate solution.

Now, let's solve the equation 2 * 2^2 * 2^3 * ... * 2^x = 2^36.

We can simplify the left side of the equation by adding the exponents:

2^(1 + 2 + 3 + ... + x) = 2^36

Using the formula for the sum of an arithmetic series, we have:

2^((x(x + 1))/2) = 2^36

Since the bases are equal, we can equate the exponents:

(x(x + 1))/2 = 36

Multiplying both sides by 2, we get:

x(x + 1) = 72

Expanding the left side, we have:

x^2 + x = 72

Rearranging the equation, we have:

x^2 + x - 72 = 0

Factoring the quadratic equation, we get:

(x + 9)(x - 8) = 0

Setting each factor equal to zero, we have:

x + 9 = 0 or x - 8 = 0

Solving for x, we get:

x = -9 or x = 8

Therefore, the solutions to the equation 2 * 2^2 * 2^3 * ... * 2^x = 2^36 are x = -9 and x = 8.