b) $$ \left(\frac{(3 x)^{-2}}{3 x^{-2}}\right)^{-\frac{2}{3}} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
\left(\frac{(3x)^{-2}}{3x^{-2}}\right)^{-\frac{2}{3}}
$$

Let's break it down step by step.

### Step 1: Simplify the Inner Fraction

First, simplify the fraction inside the parentheses:

$$
\frac{(3x)^{-2}}{3x^{-2}}
$$

**Breakdown:**

1. **Expand $$(3x)^{-2}$$:**
   $$
   (3x)^{-2} = 3^{-2} \cdot x^{-2}
   $$
   
2. **Rewrite the denominator $$3x^{-2}$$:**
   $$
   3x^{-2} = 3^1 \cdot x^{-2}
   $$

3. **Form the fraction:**
   $$
   \frac{3^{-2} \cdot x^{-2}}{3^1 \cdot x^{-2}} = \frac{3^{-2}}{3^1} \cdot \frac{x^{-2}}{x^{-2}}
   $$

4. **Simplify the $$x$$ terms:**
   $$
   \frac{x^{-2}}{x^{-2}} = 1
   $$

5. **Simplify the constants:**
   $$
   \frac{3^{-2}}{3^1} = 3^{-2 - 1} = 3^{-3}
   $$

So, the inner fraction simplifies to:

$$
3^{-3}
$$

### Step 2: Apply the Outer Exponent

Now, apply the exponent $$-\frac{2}{3}$$ to the simplified fraction:

$$
\left(3^{-3}\right)^{-\frac{2}{3}}
$$

**Using the Power of a Power Property $$(a^m)^n = a^{mn}$$:**

$$
3^{-3 \cdot \left(-\frac{2}{3}\right)} = 3^{2} = 9
$$

### Final Answer

$$
9
$$
The simplified result is 9.

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