What is the value of z in 1/4 raised to the 3z-1 = 16 raised to the z+2 times 64 raised to the z-2

  3/8

Step-by-step explanation:

In terms of powers of 4, we have ...

  (4^-1)^(3z -1) = (4^2)^(z+2)×(4^3)^(z-2)

Taking logarithms base 4*, we have ...

  -(3z -1) = 2(z +2) +3(z -2)

  -3z +1 = 5z -2 . . . . simplify

  3 = 8z . . . . . . . . . . add 3z+2

  z = 3/8 . . . . . . . . . .divide by 8

* This is equivalent to equating exponents of the same base. You're making use of the rules of exponents ...

  (a^b)^c = a^(bc)

  (a^b)(a^c) = a^(b+c)

  1/a = a^-1

and the logarithm relations ...

  logb(b^x) = x

  log(xy) = log(x) +log(y)

25/14 or 1.

step-by-step explanation:

the dots mean that it has infinitely more numbers behind it