https://r2.erweima.ai/mathbot/image/17266984468756e0oj0fd.jpg

Let's analyze and clarify the given equation. Here's the breakdown from the image you've provided:

The equation in question is:

\[
\left(\frac{1}{81}\right)^{-x} = (9)^{2x} = (27)^{3x - 2}
\]

### Step 1: Express the powers in terms of a common base

We know the following:
- \(81 = 3^4\)
- \(9 = 3^2\)
- \(27 = 3^3\)

Thus, we can rewrite the equation:

\[
\left(\frac{1}{81}\right)^{-x} = (3^4)^{-x} = 3^{-4x}
\]
\[
(9)^{2x} = (3^2)^{2x} = 3^{4x}
\]
\[
(27)^{3x - 2} = (3^3)^{3x - 2} = 3^{9x - 6}
\]

So now the equation becomes:
\[
3^{-4x} = 3^{4x} = 3^{9x - 6}
\]

### Step 2: Solve the equation

Now, you can compare the exponents, as the bases are all the same:

From the first two terms:
\[
-4x = 4x
\]
This implies that \(x = 0\).

From the second and third terms:
\[
4x = 9x - 6
\]
Solve for \(x\):
\[
4x - 9x = -6
\]
\[
-5x = -6
\]
\[
x = \frac{6}{5}
\]

### Step 3: Check for consistency
We obtained \(x = 0\) from one part of the equation and \(x = \frac{6}{5}\) from another part. Since \(x = 0\) leads to an inconsistency, we discard it. The solution to the equation is:

\[
x = \frac{6}{5}
\]

Let me know if you'd like further details on any part or have any questions!

### Here are five related questions you can explore:
1. What are the rules for simplifying exponents with negative bases?
2. How can you solve exponential equations with different bases?
3. What is the significance of the common base approach when solving equations?
4. Can \(x = 0\) be a valid solution in any variation of this equation?
5. How do you check the consistency of multiple solutions in equations with exponents?

### Tip: 
Always remember to first express all terms with the same base in exponential equations. This will allow you to equate the exponents and simplify the problem effectively.

Help me solve the equation involving exponents and simplify it.

Discover how to solve exponential equations by converting all terms to a common base and using the properties of exponents. This page provides a step-by-step explanation to help students understand and apply these techniques effectively. Suitable for high school students studying algebra.

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