Question
2. Simplify: \( \begin{array}{ll}\text { a) } \frac{2^{n+1}+2^{n+3}}{2^{n+1}} & \text { b) } \frac{3^{x+2}-3^{x}}{3^{x+1}+3^{x}}\end{array} \)
Ask by Mccoy Goodwin. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
a) 5
b) 2
Solution
Let's simplify each expression step by step.
### a) \( \frac{2^{n+1} + 2^{n+3}}{2^{n+1}} \)
1. **Factor out \( 2^{n+1} \) in the numerator:**
\[
\frac{2^{n+1} (1 + 2^{2})}{2^{n+1}}
\]
2. **Simplify \( 2^2 \):**
\[
\frac{2^{n+1} (1 + 4)}{2^{n+1}} = \frac{2^{n+1} \times 5}{2^{n+1}}
\]
3. **Cancel out \( 2^{n+1} \):**
\[
5
\]
**Simplified Result:** \( 5 \)
---
### b) \( \frac{3^{x+2} - 3^{x}}{3^{x+1} + 3^{x}} \)
1. **Factor out \( 3^{x} \) in the numerator and denominator:**
\[
\frac{3^{x} (3^{2} - 1)}{3^{x} (3^{1} + 1)}
\]
2. **Simplify the exponents:**
\[
\frac{3^{x} (9 - 1)}{3^{x} (3 + 1)} = \frac{3^{x} \times 8}{3^{x} \times 4}
\]
3. **Cancel out \( 3^{x} \):**
\[
\frac{8}{4} = 2
\]
**Simplified Result:** \( 2 \)
---
**Final Answers:**
- **a)** \( 5 \)
- **b)** \( 2 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
For part a), you can factor out \(2^{n+1}\) from the numerator: \[ \frac{2^{n+1}(1 + 2^2)}{2^{n+1}} = 1 + 4 = 5. \] So the simplified form is 5. For part b), factor \(3^{x}\) from the numerator and denominator: \[ \frac{3^{x}(3^2 - 1)}{3^{x}(3 + 1)} = \frac{9 - 1}{3 + 1} = \frac{8}{4} = 2. \] Thus, the simplified expression is 2.
