To solve the equation \( 16^{x-7} = 256 \), follow these steps:

### Step 1: Express both sides as powers of the same base.

We know that:
\[
16 = 2^4 \quad \text{and} \quad 256 = 2^8
\]
So, rewrite the equation:
\[
(2^4)^{x-7} = 2^8
\]

### Step 2: Apply the power rule \((a^m)^n = a^{m \cdot n}\).

Using this rule, we simplify the left side:
\[
2^{4(x-7)} = 2^8
\]

### Step 3: Set the exponents equal to each other.

Since the bases are the same, the exponents must also be equal:
\[
4(x-7) = 8
\]

### Step 4: Solve for \( x \).

First, distribute the 4 on the left-hand side:
\[
4x - 28 = 8
\]

Add 28 to both sides:
\[
4x = 36
\]

Now, divide both sides by 4:
\[
x = 9
\]

### Final Answer:
\[
x = 9
\]

Would you like a detailed explanation or have any questions?

Here are 5 related questions:
1. How do you simplify exponential expressions?
2. What is the power rule in exponents?
3. Can you explain how to solve equations with logarithms?
4. How do you solve exponential equations with different bases?
5. How can you check if your solution to an equation is correct?

**Tip:** When solving exponential equations, always try to express both sides as powers of the same base. This simplifies the process of equating the exponents.

Learn how to solve the exponential equation 16^(x-7) = 256 using exponent rules. This step-by-step solution includes expressing both sides as powers of 2 and equating the exponents. Suitable for students in Grades 8-10.

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