Certainly! Let's solve the given equation step by step.
The equation to solve is:
[tex]\[
\left(\frac{1}{4}\right)^{3z - 1} = 16^{z + 2} \cdot 64^{z - 2}
\][/tex]
First, let's express all terms with the same base.
### Step 1: Simplify the Right Side
We know that:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ 64 = 2^6 \][/tex]
So, we can rewrite the right-hand side as:
[tex]\[
16^{z + 2} = (2^4)^{z + 2} = 2^{4(z + 2)}
\][/tex]
[tex]\[
64^{z - 2} = (2^6)^{z - 2} = 2^{6(z - 2)}
\][/tex]
Now let's combine these:
[tex]\[
2^{4(z + 2)} \cdot 2^{6(z - 2)}
\][/tex]
### Step 2: Apply the properties of exponents
When multiplying exponents with the same base, we add the exponents:
[tex]\[
2^{4(z + 2) + 6(z - 2)}
\][/tex]
Simplify the exponent:
[tex]\[
4(z + 2) + 6(z - 2) = 4z + 8 + 6z - 12 = 10z - 4
\][/tex]
So the right side becomes:
[tex]\[
2^{10z - 4}
\][/tex]
### Step 3: Simplify the Left Side
Note that:
[tex]\[
\frac{1}{4} = 4^{-1} \quad \text{and} \quad 4 = 2^2
\][/tex]
So:
[tex]\[
\left(\frac{1}{4}\right)^{3z - 1} = \left(4^{-1}\right)^{3z - 1} = 4^{-(3z - 1)} = (2^2)^{-(3z - 1)} = 2^{-2(3z - 1)}
\][/tex]
Simplify this exponent:
[tex]\[
2^{-2(3z - 1)} = 2^{-6z + 2}
\][/tex]
### Step 4: Set the Exponents Equal
Now we have:
[tex]\[
2^{-6z + 2} = 2^{10z - 4}
\][/tex]
Since the bases are the same, the exponents must be equal:
[tex]\[
-6z + 2 = 10z - 4
\][/tex]
### Step 5: Solve for [tex]\( z \)[/tex]
Combine like terms and solve for [tex]\( z \)[/tex]:
[tex]\[
2 + 4 = 10z + 6z
\][/tex]
[tex]\[
6 = 16z
\][/tex]
[tex]\[
z = \frac{6}{16} = \frac{3}{8}
\][/tex]
So the solution is:
[tex]\[
z = \frac{3}{8}
\][/tex]
