The given equation is:

$2^{4x - 9} = \left( \frac{1}{8} \right)^{x - 4}$

Step-by-Step Solution:

1. Rewrite $\frac{1}{8}$ as a power of 2: $\frac{1}{8} = 8^{-1} = (2^3)^{-1} = 2^{-3}$ So the equation becomes: $2^{4x - 9} = (2^{-3})^{x - 4}$

2. Simplify the right-hand side using exponent rules: $(2^{-3})^{x - 4} = 2^{-3(x - 4)} = 2^{-3x + 12}$

3. Now the equation is: $2^{4x - 9} = 2^{-3x + 12}$ Since the bases are the same, we can set the exponents equal to each other: $4x - 9 = -3x + 12$

4. Solve for $x$: $4x + 3x = 12 + 9$ $7x = 21$ $x = 3$

Thus, the solution is $x = 3$.

Final Answer:

The solution is $x = 3.0$, written as a decimal number.

Would you like further details or explanations?

Related Questions:

1. How would the solution change if the exponent on the left-hand side was different?
2. Can you generalize this method to solve equations of the form $a^{bx + c} = d^{ex + f}$?
3. What happens if the base is not the same on both sides of the equation?
4. How would the solution change if fractions were involved in both exponents?
5. What are some other properties of exponents that help in solving similar equations?

Tip:

Always try to express both sides of an equation with the same base when solving exponential equations—it simplifies the problem greatly!