Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( \frac{3^{2x-1}-3^{x-1}-2}{3^{x-1}-1} \).
Solution
Calculate or simplify the expression \( (3^(2*x)-3^(x)-6)/(3^(x)-3) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(3^{2x}-3^{x}-6\right)}{\left(3^{x}-3\right)}\)
- step1: Remove the parentheses:
\(\frac{3^{2x}-3^{x}-6}{3^{x}-3}\)
- step2: Calculate:
\(\frac{\left(3^{2x-1}-3^{x-1}-2\right)\times 3}{\left(3^{x-1}-1\right)\times 3}\)
- step3: Reduce the fraction:
\(\frac{3^{2x-1}-3^{x-1}-2}{3^{x-1}-1}\)
The simplified form of the expression \( \frac{3^{2x}-3^{x}-6}{3^{x}-3} \) is \( \frac{3^{2x-1}-3^{x-1}-2}{3^{x-1}-1} \).
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Simplify this solution Beyond the Answer
To simplify the expression \( \frac{3^{2x}-3^{x}-6}{3^{x}-3} \), let's start by making a substitution. Let \( y = 3^{x} \). Then, we can write \( 3^{2x} = (3^{x})^2 = y^2 \). Now we can rewrite the expression: \[ \frac{y^2 - y - 6}{y - 3} \] Next, we can factor the numerator: \[ y^2 - y - 6 = (y - 3)(y + 2) \] So the expression becomes: \[ \frac{(y - 3)(y + 2)}{y - 3} \] Since \( y \neq 3 \) (we will avoid this value to prevent division by zero), we can cancel \( y - 3 \): \[ y + 2 \] Now substituting back \( y = 3^{x} \): \[ 3^{x} + 2 \] Thus, the simplified expression is: \[ 3^{x} + 2 \]
