c) $$ \frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2^{x} + 2^{x+1}}{2^{x} - 2^{x+2}}
$$

**Step 1: Factor out the common term $$2^{x}$$ from both the numerator and the denominator.**

- **Numerator:**
  $$
  2^{x} + 2^{x+1} = 2^{x} (1 + 2^1) = 2^{x} (1 + 2) = 2^{x} \times 3
  $$

- **Denominator:**
  $$
  2^{x} - 2^{x+2} = 2^{x} (1 - 2^{2}) = 2^{x} (1 - 4) = 2^{x} \times (-3)
  $$

**Step 2: Rewrite the original expression with the factored terms.**

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

**Step 3: Simplify the expression by canceling out the common terms.**

- The $$2^{x}$$ in the numerator and the denominator cancel each other out.
- The $$3$$ and $$-3$$ simplify as follows:

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

**Final Simplified Result:**

$$
-1
$$
The expression simplifies to $$-1$$.

c) \( \frac{2^{x}+2^{x+1}}{2^{x}-2^{x+2}} \)