Question
Solve for x . Write your answer as a decimal. \( 9^{x-7}=27 \) \( x= \)
Ask by Dawson Alexander. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 8.5 \)
Solution
To solve for \( x \) in the equation \( 9^{x-7} = 27 \), follow these steps:
1. **Rewrite the Bases as Powers of 3**:
- \( 9 \) can be written as \( 3^2 \).
- \( 27 \) can be written as \( 3^3 \).
So, the equation becomes:
\[
(3^2)^{x-7} = 3^3
\]
2. **Simplify the Left Side Using the Power of a Power Rule**:
\[
3^{2(x-7)} = 3^3
\]
3. **Set the Exponents Equal to Each Other** (since the bases are the same):
\[
2(x - 7) = 3
\]
4. **Solve for \( x \)**:
\[
2x - 14 = 3 \\
2x = 17 \\
x = \frac{17}{2} \\
x = 8.5
\]
**Answer:**
\( x = 8.5 \)
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Simplify this solution Beyond the Answer
To solve the equation \( 9^{x-7} = 27 \), we can rewrite both sides in terms of powers of 3. Notably, \( 9 \) can be expressed as \( 3^2 \) and \( 27 \) as \( 3^3 \). This gives us: \[ (3^2)^{x-7} = 3^3 \] Using the exponent rule \( (a^m)^n = a^{mn} \), we transform the left side: \[ 3^{2(x-7)} = 3^3 \] Now that the bases are the same, we can set the exponents equal to each other: \[ 2(x-7) = 3 \] Distributing the 2: \[ 2x - 14 = 3 \] Adding 14 to both sides: \[ 2x = 17 \] Finally, dividing by 2: \[ x = \frac{17}{2} = 8.5 \] So, \( x = 8.5 \)
