Question
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Answer
The simplified expression is 6.
Solution
We start with the expression:
(3^(n+1) - 3^n) / 3^(n-1)
Step 1: Factor out 3^n from the numerator:
3^(n+1) - 3^n = 3^n · (3 - 1) = 3^n · 2 = 2·3^n
Step 2: Write the expression:
(2·3^n) / 3^(n-1)
Step 3: Simplify by dividing exponentials with the same base:
3^n / 3^(n-1) = 3^(n - (n-1)) = 3^1 = 3
Step 4: Multiply:
2 · 3 = 6
Thus, the simplified expression is 6.
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Simplify this solution The Deep Dive
To simplify the expression \( \frac{3^{n+1}-3^{n}}{3^{n-1}} \), let’s first factor out the numerator. Notice that \( 3^{n+1} - 3^{n} = 3^{n}(3 - 1) = 3^{n} \cdot 2 \). Now we can rewrite the expression as: \[ \frac{3^{n} \cdot 2}{3^{n-1}} \] Next, we can simplify further by canceling \( 3^{n} \) in the numerator with \( 3^{n-1} \) in the denominator: \[ = 2 \cdot \frac{3^{n}}{3^{n-1}} = 2 \cdot 3^{1} = 2 \cdot 3 = 6 \] So, the simplified expression is \( 6 \).
