Pregunta
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\( x = \frac{1}{4} \)
Solución
To solve the equation \( 2^{-3x} = 2^{x - 1} \), we can use the property of exponents that states if \( 2^a = 2^b \), then \( a = b \). Here's a step-by-step solution:
1. **Start with the given equation:**
\[
2^{-3x} = 2^{x - 1}
\]
2. **Since the bases are the same (both are 2), set the exponents equal to each other:**
\[
-3x = x - 1
\]
3. **Solve for \( x \):**
\[
-3x = x - 1 \\
-3x - x = -1 \quad \text{(Subtract \( x \) from both sides)} \\
-4x = -1 \\
x = \frac{-1}{-4} \\
x = \frac{1}{4}
\]
4. **Conclusion:**
\[
x = \frac{1}{4}
\]
**Verification:**
To ensure the solution is correct, substitute \( x = \frac{1}{4} \) back into the original equation:
\[
2^{-3(\frac{1}{4})} = 2^{\frac{1}{4} - 1} \\
2^{-\frac{3}{4}} = 2^{-\frac{3}{4}}
\]
Both sides are equal, confirming that \( x = \frac{1}{4} \) is the correct solution.
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución The Deep Dive
To solve the equation \( 2^{-3x} = 2^{x-1} \), you can equate the exponents since the bases (2 in this case) are the same. This gives you: \[-3x = x - 1.\] Now, let's solve for \(x\). First, add \(3x\) to both sides: \[0 = 4x - 1.\] Next, add 1 to both sides: \[1 = 4x.\] Finally, divide by 4 to isolate \(x\): \[x = \frac{1}{4}.\] So the solution to the equation is \(x = \frac{1}{4}\).
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