Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( n = -3 \)
Solution
To solve the equation \( 6^{-2n - 3} = 216 \) for \( n \), follow these steps:
1. **Recognize the Base and Exponent:**
The equation involves exponentials with base 6. Notice that \( 216 \) is a power of 6:
\[
6^3 = 6 \times 6 \times 6 = 216
\]
2. **Rewrite the Equation with the Same Base:**
Replace \( 216 \) with \( 6^3 \):
\[
6^{-2n - 3} = 6^3
\]
3. **Set the Exponents Equal:**
Since the bases are the same and the equation holds true, their exponents must be equal:
\[
-2n - 3 = 3
\]
4. **Solve for \( n \):**
- Add 3 to both sides to isolate the term with \( n \):
\[
-2n = 3 + 3 \\
-2n = 6
\]
- Divide both sides by -2 to solve for \( n \):
\[
n = \frac{6}{-2} \\
n = -3
\]
**Final Answer:**
\[
n = -3
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the equation \( 6^{-2n-3} = 216 \), first express 216 as a power of 6. Since \( 6^3 = 216 \), we can rewrite the equation as \( 6^{-2n-3} = 6^3 \). This means the exponents must be equal, so we set up the equation: \[ -2n - 3 = 3 \] Now, add 3 to both sides: \[ -2n = 6 \] Then, divide by -2: \[ n = -3 \] Therefore, the solution is \( n = -3 \).
