The answer is b = -6.
You can find this fairly simply. First, distribute the power of the outside of the parenthesis into it.
[tex] y^{4b} = \frac{1}{y^{24}} [/tex]
Now that we have this, we can manipulate the right side of the equation. When a power is on the bottom of an equation, we can move it to the top if we make it negative.
[tex] y^{4b} = y^{-24} [/tex]
Now since we have the same base, in order for them to be equal, we would need for the powers to be equal. This means we can set them equal to each other and solve.
4b = -24
b = -6
