To solve the equation [tex]\(\left(\frac{1}{4}\right)^{3z-1} = 16^{z+2} \cdot 64^{z-2}\)[/tex], let's proceed through the following steps:
1. Rewrite all terms with the same base:
- Notice that [tex]\(\frac{1}{4} = 4^{-1}\)[/tex].
- Note that [tex]\(16 = 4^2\)[/tex].
- Also, [tex]\(64 = 4^3\)[/tex].
Thus, we can rewrite the original equation:
[tex]\[
\left(4^{-1}\right)^{3z-1} = \left(4^2\right)^{z+2} \cdot \left(4^3\right)^{z-2}
\][/tex]
2. Simplify the exponents using properties of exponents:
- For the left-hand side:
[tex]\[
\left(4^{-1}\right)^{3z-1} = 4^{(-1)(3z-1)} = 4^{-(3z-1)}
\][/tex]
- For the right-hand side:
[tex]\[
\left(4^2\right)^{z+2} = 4^{2(z+2)} = 4^{2z+4}
\][/tex]
[tex]\[
\left(4^3\right)^{z-2} = 4^{3(z-2)} = 4^{3z-6}
\][/tex]
- Combining exponents when the bases are the same on the right-hand side:
[tex]\[
4^{2z+4} \cdot 4^{3z-6} = 4^{(2z+4) + (3z-6)} = 4^{2z + 4 + 3z - 6} = 4^{5z - 2}
\][/tex]
Thus, the equation simplifies to:
[tex]\[
4^{-(3z-1)} = 4^{5z-2}
\][/tex]
3. Set the exponents equal to each other:
Since the bases are the same and the equation holds, we can set the exponents equal:
[tex]\[
-(3z-1) = 5z-2
\][/tex]
4. Solve the resulting equation:
[tex]\[
-(3z-1) = 5z-2
\][/tex]
Distribute the negative sign:
[tex]\[
-3z + 1 = 5z - 2
\][/tex]
Combine like terms by adding [tex]\(3z\)[/tex] to both sides:
[tex]\[
1 = 8z - 2
\][/tex]
Add 2 to both sides to isolate terms involving [tex]\(z\)[/tex]:
[tex]\[
3 = 8z
\][/tex]
Divide by 8:
[tex]\[
z = \frac{3}{8}
\][/tex]
Therefore, the solution to the equation [tex]\(\left(\frac{1}{4}\right)^{3z-1} = 16^{z+2} \cdot 64^{z-2}\)[/tex] is:
[tex]\[
z = \frac{3}{8}
\][/tex]
