Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( \frac{25}{4} \).
Solution
To simplify the expression:
\[
\frac{7^{x} + 7^{x+2}}{7^{x+1} + 7^{x}}
\]
**Step 1: Factor Out Common Terms**
- **Numerator:**
\[
7^{x} + 7^{x+2} = 7^{x} \left(1 + 7^{2}\right) = 7^{x} \times 50
\]
- **Denominator:**
\[
7^{x+1} + 7^{x} = 7^{x} \left(7 + 1\right) = 7^{x} \times 8
\]
**Step 2: Simplify the Fraction**
\[
\frac{7^{x} \times 50}{7^{x} \times 8} = \frac{50}{8} = \frac{25}{4}
\]
**Final Answer:**
\[
\frac{25}{4}
\]
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Simplify this solution Mind Expander
To simplify the expression \( \frac{7^{x}+7^{x+2}}{7^{x+1}+7^{x}} \), we can factor out \( 7^{x} \) from both the numerator and the denominator. Starting with the numerator: \[ 7^{x} + 7^{x+2} = 7^{x} + 7^{x} \cdot 7^2 = 7^{x}(1 + 49) = 7^{x} \cdot 50 \] Now, for the denominator: \[ 7^{x+1} + 7^{x} = 7^{x} \cdot 7 + 7^{x} = 7^{x}(7 + 1) = 7^{x} \cdot 8 \] Now we can rewrite the original expression using these simplifications: \[ \frac{7^{x}(50)}{7^{x}(8)} \] This allows us to cancel out \( 7^{x} \) from the numerator and the denominator (assuming \( 7^{x} \neq 0 \)): \[ \frac{50}{8} = \frac{25}{4} \] Thus, the simplified form of the original expression is: \[ \frac{25}{4} \]
