https://file.aiquickdraw.com/mathbot/file/1729898554842qdmffdq2.jpg

Let's solve the given equation:

\[
16^{x-2} = 32^{x+3}
\]

### Step 1: Express the bases as powers of 2

We know that:
\[
16 = 2^4 \quad \text{and} \quad 32 = 2^5
\]

Thus, we can rewrite the equation as:

\[
(2^4)^{x-2} = (2^5)^{x+3}
\]

### Step 2: Apply the power of a power rule

The power of a power rule states \((a^m)^n = a^{m \cdot n}\). Using this rule, we get:

\[
2^{4(x-2)} = 2^{5(x+3)}
\]

### Step 3: Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other:

\[
4(x - 2) = 5(x + 3)
\]

### Step 4: Expand both sides

\[
4x - 8 = 5x + 15
\]

### Step 5: Solve for \(x\)

Move all terms involving \(x\) to one side and constants to the other side:

\[
4x - 5x = 15 + 8
\]

Simplify:

\[
-x = 23
\]

Divide by \(-1\):

\[
x = -23
\]

### Final Answer:
\[
x = -23
\]

Would you like further details or have any questions? Here are five related questions to explore:

1. What if the base of 16 and 32 were different powers?
2. Can the solution change if we alter the exponents by a constant?
3. How does the power of a power rule work in fractional exponents?
4. What happens if we change the bases from 2 to another prime number?
5. How can logarithms be used to solve equations like this?

**Tip:** When solving exponential equations, converting bases to a common value often simplifies the process significantly.

This step-by-step solution guides you through solving the exponential equation 16^(x-2) = 32^(x+3). Learn how to apply the power of a power rule and simplify exponential expressions by expressing bases as powers of 2. Suitable for high school students in Grades 9-12.

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