Pregunta
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\( x = 4 \)
Solución
To solve the equation \( 4^{x} = 256 \), follow these steps:
1. **Express Both Sides with the Same Base:**
Notice that both 4 and 256 can be expressed as powers of 2.
- \( 4 = 2^2 \)
- \( 256 = 2^8 \)
2. **Rewrite the Equation:**
Substitute the expressions with base 2 into the original equation:
\[
(2^2)^x = 2^8
\]
3. **Simplify the Left Side:**
Use the exponentiation rule \((a^m)^n = a^{m \cdot n}\):
\[
2^{2x} = 2^8
\]
4. **Set the Exponents Equal:**
Since the bases are the same, the exponents must be equal:
\[
2x = 8
\]
5. **Solve for \( x \):**
\[
x = \frac{8}{2} = 4
\]
**Answer:** \( x = 4 \)
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Mind Expander
To solve the equation \( 4^{x} = 256 \), first notice that both sides can be expressed as powers of 2. Since \( 4 = 2^{2} \), we can rewrite the left side as \( (2^{2})^{x} = 2^{2x} \). The right side, \( 256 \), can also be expressed as \( 2^{8} \), because \( 2^{8} = 256 \). Now we have the equation \( 2^{2x} = 2^{8} \). By setting the exponents equal to each other, we get \( 2x = 8 \). Dividing both sides by 2 gives \( x = 4 \). So, the solution to the equation is \( x = 4 \).
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