4.2. $$ \frac{2^{2 x-1} \cdot 4^{x+1} \cdot 2}{16^{x}} $$

Question

UpStudy AI · Accepted Answer

We start with the expression:
$$
\frac{2^{2x-1} \cdot 4^{x+1} \cdot 2}{16^{x}}
$$

**Step 1. Rewrite all terms with base $$2$$.**

- $$4 = 2^2$$ and $$16 = 2^4$$.

- Rewrite $$4^{x+1}$$:
  $$
  4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}.
  $$

- Rewrite $$16^{x}$$:
  $$
  16^{x} = (2^4)^{x} = 2^{4x}.
  $$

**Step 2. Substitute the rewritten terms.**

The expression becomes:
$$
\frac{2^{2x-1} \cdot 2^{2x+2} \cdot 2}{2^{4x}}.
$$

Recall that $$2 = 2^1$$.

**Step 3. Combine the powers in the numerator.**

Add the exponents:
$$
2^{2x-1} \cdot 2^{2x+2} \cdot 2^1 = 2^{(2x-1) + (2x+2) + 1}.
$$

Simplify the exponent:
$$
(2x-1) + (2x+2) + 1 = 2x + 2x + (-1+2+1) = 4x + 2.
$$

So the numerator simplifies to:
$$
2^{4x+2}.
$$

**Step 4. Divide the powers with the same base.**

Divide the numerator by the denominator:
$$
\frac{2^{4x+2}}{2^{4x}} = 2^{(4x+2) - 4x} = 2^2.
$$

**Step 5. Simplify the final result.**

$$
2^2 = 4.
$$

Thus, the simplified expression is:
$$
4.
$$
The simplified expression is 4.

4.2.