2y^-3 = 4e^(-x/2)/(1+e(-x/2))

my teacher gave this problem in my chem class and i don't know what he is really asking , does any one have an idea if i have to solve for y or what's going on.. how would this be done..

To better understand what your teacher is asking for, let's break down the equation:

2y^-3 = 4e^(-x/2) / (1+e^(-x/2))

Here's what each part of the equation means:

- The variable "y" is raised to the power of -3, which means we have to take the reciprocal of y cubed.
- The term on the right side consists of two parts: 4e^(-x/2) and 1+e^(-x/2), both of which involve the exponential function e^(-x/2).

Based on this information, it seems that your teacher wants you to solve for the value of y in terms of x. So, we will attempt to isolate y on one side of the equation.

Step 1: Let's start by getting rid of the fraction. We can do this by multiplying both sides of the equation by the denominator, (1 + e^(-x/2)):

(1 + e^(-x/2)) * (2y^-3) = 4e^(-x/2)

This simplifies to:

2y^-3 * (1 + e^(-x/2)) = 4e^(-x/2)

Step 2: Distribute the terms on the left side of the equation:

2y^-3 + 2y^-3 * e^(-x/2) = 4e^(-x/2)

Step 3: Now, let's isolate the term with y by subtracting 2y^-3 from both sides:

2y^-3 * e^(-x/2) = 4e^(-x/2) - 2y^-3

Step 4: To further simplify the equation, let's combine like terms and put everything on one side:

2y^-3 * e^(-x/2) + 2y^-3 = 4e^(-x/2)

2y^-3 * e^(-x/2) + 2y^-3 - 4e^(-x/2) = 0

Step 5: At this point, we have a quadratic equation in terms of y^-3. Let's substitute a variable, such as "z," for y^-3 to make our calculations easier:

2z * e^(-x/2) + 2z - 4e^(-x/2) = 0

Step 6: Rearrange the terms:

2z * e^(-x/2) - 4e^(-x/2) + 2z = 0

Step 7: Factor out e^(-x/2):

2z * e^(-x/2) - 2 * e^(-x/2) + 2z = 0

Step 8: Factor out 2z from the first and last term:

2z * (e^(-x/2) + 1) - 2 * e^(-x/2) = 0

Step 9: Divide both sides of the equation by (e^(-x/2) + 1):

2z = (2 * e^(-x/2)) / (e^(-x/2) + 1)

Step 10: Simplify the right side of the equation:

2z = 2e^(-x/2) / (e^(-x/2) + 1)

Step 11: Divide both sides of the equation by 2:

z = e^(-x/2) / (e^(-x/2) + 1)

Step 12: Substitute back z for y^-3:

y^-3 = e^(-x/2) / (e^(-x/2) + 1)

Step 13: Take the reciprocal of both sides:

y^3 = (e^(-x/2) + 1) / e^(-x/2)

Step 14: Simplify the right side by dividing both terms by e^(-x/2):

y^3 = (e^(-x/2) / e^(-x/2) + 1 / e^(-x/2))

y^3 = (1 + e^(x/2))

Step 15: Finally, take the cube root of both sides to solve for y:

y = (1 + e^(x/2))^(1/3)

So, the solution for y in terms of x is y = (1 + e^(x/2))^(1/3).