Activity 1 $$ \frac{(7)\left(2^{x}\right)^{3} \cdot 2^{x}}{16^{x}} $$ $$ \frac{\left(22^{x}\right)^{3} \cdot 27^{x}}{8^{x} \cdot\left(3^{2}\right)^{x} \cdot 3^{x}} $$ (3) $$ \left(\frac{1}{4}\right)^{-3 x} \frac{1}{16^{x}} $$

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Let's simplify each expression step by step.

### Expression 1: 
$$
\frac{(7)(2^{x})^{3} \cdot 2^{x}}{16^{x}}
$$

1. **Simplify the numerator:**
   $$
   (2^{x})^{3} = 2^{3x}
   $$
   Therefore, the numerator becomes:
   $$
   7 \cdot 2^{3x} \cdot 2^{x} = 7 \cdot 2^{3x + x} = 7 \cdot 2^{4x}
   $$

2. **Simplify the denominator:**
   $$
   16^{x} = (2^{4})^{x} = 2^{4x}
   $$

3. **Combine the expression:**
   $$
   \frac{7 \cdot 2^{4x}}{2^{4x}} = 7
   $$

### Expression 2:
$$
\frac{(22^{x})^{3} \cdot 27^{x}}{8^{x} \cdot (3^{2})^{x} \cdot 3^{x}}
$$

1. **Simplify the numerator:**
   $$
   (22^{x})^{3} = 22^{3x}
   $$
   Therefore, the numerator becomes:
   $$
   22^{3x} \cdot 27^{x}
   $$

2. **Simplify the denominator:**
   $$
   8^{x} = (2^{3})^{x} = 2^{3x}
   $$
   $$
   (3^{2})^{x} = 3^{2x}
   $$
   Therefore, the denominator becomes:
   $$
   2^{3x} \cdot 3^{2x} \cdot 3^{x} = 2^{3x} \cdot 3^{3x}
   $$

3. **Combine the expression:**
   $$
   \frac{22^{3x} \cdot 27^{x}}{2^{3x} \cdot 3^{3x}}
   $$

Now, we can express $$27^{x}$$ as $$3^{3x}$$:
   $$
   \frac{22^{3x} \cdot 3^{3x}}{2^{3x} \cdot 3^{3x}} = \frac{22^{3x}}{2^{3x}} = \left(\frac{22}{2}\right)^{3x} = 11^{3x}
   $$

### Expression 3:
$$
\left(\frac{1}{4}\right)^{-3x} \cdot \frac{1}{16^{x}}
$$

1. **Simplify the first part:**
   $$
   \left(\frac{1}{4}\right)^{-3x} = 4^{3x} = (2^{2})^{3x} = 2^{6x}
   $$

2. **Simplify the second part:**
   $$
   16^{x} = (2^{4})^{x} = 2^{4x}
   $$

3. **Combine the expression:**
   $$
   2^{6x} \cdot \frac{1}{2^{4x}} = 2^{6x - 4x} = 2^{2x}
   $$

### Final Results:
1. The first expression simplifies to $$7$$.
2. The second expression simplifies to $$11^{3x}$$.
3. The third expression simplifies to $$2^{2x}$$.

Thus, the simplified forms of the expressions are:
1. $$7$$
2. $$11^{3x}$$
3. $$2^{2x}$$
The simplified forms of the expressions are: 1. 7 2. $$11^{3x}$$ 3. $$2^{2x}$$

Activity 1 \( \frac{(7)\left(2^{x}\right)^{3} \cdot 2^{x}}{16^{x}} \) \( \frac{\left(22^{x}\right)^{3} \cdot 27^{x}}{8^{x} \cdot\left(3^{2}\right)^{x} \cdot 3^{x}} \) (3) \( \left(\frac{1}{4}\right)^{-3 x} \frac{1}{16^{x}} \)

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