b.) $$ \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2^{x+2} - 2^{x+3}}{2^{x+1} - 2^{x+2}}
$$

**Step 1: Factor Out Common Terms**

Start by factoring out the smallest power of 2 in both the numerator and the denominator.

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

- **Denominator:**
  $$
  2^{x+1} - 2^{x+2} = 2^{x+1}(1 - 2) = 2^{x+1}(-1) = -2^{x+1}
  $$

**Step 2: Simplify the Expression**

Now, substitute the factored forms back into the original expression:

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

**Step 3: Use the Laws of Exponents**

When dividing like bases, subtract the exponents:

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

**Final Answer:**  
$$
2
$$
The simplified form of the expression is 2.

b.) \( \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} \)