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

Question

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

UpStudy AI · Accepted Answer

Let's simplify the given expression step by step. The expression provided is:

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

**Step 1: Simplify the Numerator**

First, simplify the inner fraction in the numerator:

$$
\frac{3^{2}}{3^{4} \cdot 3^{4}} = \frac{3^{2}}{3^{8}} = 3^{2 - 8} = 3^{-6}
$$

Now, raise this result to the power of 3:

$$
\left(3^{-6}\right)^{3} = 3^{-18} = \frac{1}{3^{18}}
$$

**Step 2: Simplify the Denominator**

Next, simplify the fraction in the denominator:

$$
\frac{2^{3} \cdot 2^{2}}{2^{2}} = \frac{2^{5}}{2^{2}} = 2^{5 - 2} = 2^{3} = 8
$$

**Step 3: Combine Numerator and Denominator**

Now, divide the simplified numerator by the simplified denominator:

$$
\frac{\frac{1}{3^{18}}}{8} = \frac{1}{8 \cdot 3^{18}}
$$

**Step 4: Compute the Numerical Value**

Calculate $$3^{18}$$:

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

Multiply by 8:

$$
8 \cdot 387,420,489 = 3,099,363,912
$$

So, the final simplified form of the expression is:

$$
\frac{1}{3,099,363,912}
$$

**Final Answer:**

$$
\frac{1}{3,\!099,\!363,\!912}
$$
The simplified answer is $$ \frac{1}{3,\!099,\!363,\!912} $$.

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