https://file.aiquickdraw.com/mathbot/file/172559679787359sxa1ib.jpg

The equation provided in the image is:

\[
10^{3x} = 10,000^2
\]

Let's solve this step by step:

### Step 1: Express \( 10,000 \) as a power of 10.
We know that:
\[
10,000 = 10^4
\]
So, the equation becomes:
\[
10^{3x} = (10^4)^2
\]

### Step 2: Simplify the right-hand side.
Using the power rule \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[
(10^4)^2 = 10^{8}
\]
So, the equation now is:
\[
10^{3x} = 10^8
\]

### Step 3: Equate the exponents.
Since the bases are the same, we can set the exponents equal to each other:
\[
3x = 8
\]

### Step 4: Solve for \(x\).
\[
x = \frac{8}{3}
\]

Thus, the solution is:
\[
x = \frac{8}{3} \approx 2.67
\]

Would you like further clarification on any step? Here are some related questions to explore:

1. How can we express other large numbers as powers of 10?
2. What other rules of exponents are useful in solving equations like this?
3. Can we solve equations involving logarithms with different bases?
4. How do we apply the change of base formula for logarithms?
5. What are some common mistakes when simplifying exponentials?

**Tip**: Always check if you can simplify numbers by expressing them as powers of 10, especially when dealing with powers and exponents.

This page explains how to solve the exponential equation 10^{3x} = 10,000^2 step-by-step. We simplify powers of 10 and solve for x, focusing on exponent rules. Suitable for high school students in Grades 9-12.

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