Question

I just need the answer to check my work no explanation needed

Answer 1

To solve for z, we proceed as follows:

`((1)/(4))^(3z-1)=16^(z+2)*64^(z-2)`

Now, we simplify the expression on the right-hand side of the equation as follows:

`((1)/(4))^(3z-1)=(4^2)^(z+2)*(4^3)^(z-2)`
`((1)/(4))^(3z-1)=4^2^((z+2))*4^3^((z-2))`
`((1)/(4))^(3z-1)=4^(2z+4)*4^(3z-6)`
`((1)/(4))^(3z-1)=4^(2z+4+3z-6)`
`((1)/(4))^(3z-1)=4^(5z-2)`

Now, we simplify the expression on the left-hand side of the equation as follows:

`(4^(-1))^(3z-1)=4^(5z-2)`
`4^(-1(3z-1))=4^(5z-2)`
`4^(-3z+1)=4^(5z-2)`

Now, since we have both expressions on the left and right hand sides to have a base of 4, we can simply equate their indices, as follow:

`\begin{gathered} 4^(-3z+1)=4^(5z-2) \\ \Rightarrow-3z+1=5z-2 \\ \end{gathered}`

Now, we collect like terms:

`-3z-5z=-2-1`
`\begin{gathered} -8z=-3 \\ \Rightarrow z=(-3)/(-8) \\ \Rightarrow z=(3)/(8) \end{gathered}`