What is the quotient of the expression 3x^4 divided by 4x^2 divided by 8x divided by 1 divided by x divided by 2?

To find the quotient of the expression 3x⁴ divided by 4x² divided by 8x divided by 1 divided by x divided by 2, we need to break down this operation step by step.

1. **First Step**: Start with the first part of the expression, which is 3x⁴ divided by 4x².

The division can be expressed as:

3x4 / 4x2 = (3/4) * (x4 / x2)

When simplifying the x terms, we use the rule that states: xa / xb = xa-b. Thus,
x4 / x2 = x(4-2) = x2.

Therefore, we have:

3x4 / 4x2 = (3/4) * x2

2. **Second Step**: Now take this result and divide by 8x:

((3/4) * x2) / (8x) = (3/4) * (1/8) * (x2 / x)

When simplifying, we apply the same x division rule:
x2 / x = x(2-1) = x1 = x.

Thus, our expression simplifies to:

=(3/32) * x

3. **Third Step**: Now, we will proceed with dividing by 1 which does not affect our expression, so we still have:

=(3/32) * x

4. **Fourth Step**: Next, we divide this result by x:

((3/32) * x) / x = (3/32) * (x / x) = (3/32) * 1

Hence, this reduces to:

=(3/32)

5. **Final Step**: Lastly, we divide by 2:

(3/32) / 2 = (3/32) * (1/2) = 3/64

Thus, the final quotient of the expression is 3/64.