We have to solve 1/1024 = 4*16^2x for x.

1/1024 = 4*16^2x

=> 1/(2^10) = (2^2)*(2^4^2x)

=> 1/(2^10) = (2^2)*(2^8x)

=> [1/(2^10)]/(2^2) = (2^8x)

=> 2^(-10 - 2) = (2^2)*(2^8x)

=> 2^-12 = (2^8x)

As the base is the same we equate the exponent.

-12 = 8x

=> x = -12/8

=> x = -3/2

**The required value is x = -3/2**

We'll have to solve an exponential equation.

We notice that the denominator of the fraction from the left side could be written as:

1024 = 256*4 = 4*16^2

We'll multiply both sides by 4:

1/16^2 = 4*4*16^2x

1/16^2 = 16*16^2x

We'll re-write the right side using the property of exponentials:

a^b*a^c = a^(b+c)

16*16^2x = 16^(1+2x)

We'll re-write the equation:

1/16^2 = 16^(1+2x)

16^(-2) = 16^(1+2x)

Since the bases are matching, we'll apply one to one property of exponentials:

2x+1=-2

2x=-2-1

2x=-3

**The solution of the exponential equation 1/1024=4*16^2x is x=-3/2.**