Question: Find the solution to the following equation.
$2^{3x} = 8^{1 - x}$
Option 1: $x = \frac{1}{2}$
Option 2: $x = \frac{3}{2}$
Option 3: $x = \frac{4}{7}$
Option 4: $x = \frac{7}{4}$
1. **State the problem:** Solve the equation $$2^{3x} = 8^{1 - x}$$ for $x$.
2. **Rewrite the bases:** Note that $8$ can be written as a power of $2$ since $8 = 2^3$. So,
$$8^{1 - x} = (2^3)^{1 - x} = 2^{3(1 - x)} = 2^{3 - 3x}$$
3. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal:
$$3x = 3 - 3x$$
4. **Solve for $x$:**
$$3x + 3x = 3$$
$$6x = 3$$
$$x = \frac{3}{6} = \frac{1}{2}$$
5. **Check the options:** The solution is $x = \frac{1}{2}$, which corresponds to Option 1.
**Final answer:** $x = \frac{1}{2}$