Question
b.) \( \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} \)
Ask by Weaver Valdez. in South Africa
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is 2.
Solution
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:**
\[
2
\]
Reviewed and approved by the UpStudy tutoring team
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The Deep Dive
To simplify the expression \( \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} \), let's start with the numerator: \[ 2^{x+2} - 2^{x+3} = 2^{x+2}(1 - 2) = 2^{x+2}(-1) = -2^{x+2} \] Next, let's simplify the denominator: \[ 2^{x+1} - 2^{x+2} = 2^{x+1}(1 - 2) = 2^{x+1}(-1) = -2^{x+1} \] Now we can rewrite the entire expression: \[ \frac{-2^{x+2}}{-2^{x+1}} = \frac{2^{x+2}}{2^{x+1}} \] When we simplify this, we use the property of exponents: \[ \frac{2^{x+2}}{2^{x+1}} = 2^{(x+2)-(x+1)} = 2^{1} = 2 \] Thus, the simplified expression is: \[ \boxed{2} \]