Question
Question 1 \( 1 \cdot 1 \frac{9^{x}+3^{2 x+1}}{18^{x} \cdot 2^{-x}} \)
Ask by Fuentes Olson. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified value of the expression is 4.
Solution
To simplify the expression:
\[
\frac{9^{x} + 3^{2x + 1}}{18^{x} \cdot 2^{-x}}
\]
Let's break it down step by step.
### Step 1: Rewrite the Terms with Powers of 3 and 2
1. **Express \(9^{x}\) in terms of \(3\):**
\[
9^{x} = (3^2)^x = 3^{2x}
\]
2. **Express \(3^{2x + 1}\) as a product:**
\[
3^{2x + 1} = 3^{2x} \cdot 3^1 = 3^{2x} \cdot 3
\]
3. **Express \(18^{x}\) in terms of \(2\) and \(3\):**
\[
18^{x} = (2 \cdot 9)^x = 2^{x} \cdot 9^{x} = 2^{x} \cdot 3^{2x}
\]
4. **Express \(2^{-x}\) as a reciprocal:**
\[
2^{-x} = \frac{1}{2^{x}}
\]
### Step 2: Substitute Back into the Original Expression
\[
\frac{3^{2x} + 3^{2x} \cdot 3}{2^{x} \cdot 3^{2x} \cdot \frac{1}{2^{x}}}
\]
### Step 3: Simplify the Denominator
\[
2^{x} \cdot 3^{2x} \cdot \frac{1}{2^{x}} = 3^{2x}
\]
### Step 4: Simplify the Entire Expression
\[
\frac{3^{2x} + 3^{2x} \cdot 3}{3^{2x}} = \frac{3^{2x}(1 + 3)}{3^{2x}} = 1 + 3 = 4
\]
### Final Answer
\[
4
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To simplify the expression \( 1 \cdot 1 \frac{9^{x}+3^{2 x+1}}{18^{x} \cdot 2^{-x}} \), we start by rewriting the components: 1. Notice that \( 9^{x} = (3^2)^{x} = 3^{2x} \), and hence the expression in the numerator becomes \( 3^{2x} + 3^{2x+1} = 3^{2x}(1 + 3) = 4 \cdot 3^{2x} \). 2. In the denominator, \( 18^{x} = (2 \cdot 3^2)^{x} = 2^{x} \cdot 3^{2x} \). Since we also have \( 2^{-x} \), we can rewrite \( 18^{x} \cdot 2^{-x} \) as \( 2^{x} \cdot 3^{2x} \cdot 2^{-x} = 3^{2x} \). Now substituting these back into the simplified expression, we have: \[ \frac{4 \cdot 3^{2x}}{3^{2x}} = 4 \] Thus, the simplified answer is \( 4 \).
