2) $$ \left(\frac{3^{2}}{3^{4} \cdot 3^{4}}\right)^{3} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\left(\frac{3^{2}}{3^{4} \cdot 3^{4}}\right)^{3}
$$

Let's break it down step by step.

1. **Simplify the Denominator:**

$$
   3^{4} \cdot 3^{4} = 3^{4 + 4} = 3^{8}
   $$

2. **Divide the Numerator by the Denominator:**

$$
   \frac{3^{2}}{3^{8}} = 3^{2 - 8} = 3^{-6}
   $$

3. **Raise the Result to the Power of 3:**

$$
   \left(3^{-6}\right)^{3} = 3^{-6 \times 3} = 3^{-18}
   $$

4. **Express the Negative Exponent as a Reciprocal:**

$$
   3^{-18} = \frac{1}{3^{18}}
   $$

5. **Calculate $$3^{18}$$ (if a numerical answer is required):**

$$
   3^{18} = 387,420,489
   $$

Therefore,

$$
   \frac{1}{3^{18}} = \frac{1}{387,420,489}
   $$

**Final Answer:**

$$
\frac{1}{387,\!420,\!489}
$$
$$ \frac{1}{387,\!420,\!489} $$

Upstudy AI Solution