Let's solve each equation to find their solutions:
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Equation: \( x^4 = -1 \)
Solution: There are no real solutions because the fourth power of a real number cannot be negative. However, in the complex number system, the solutions would be \( x = e^{i\frac{\pi}{4}} \), \( e^{i\frac{3\pi}{4}} \), \( e^{i\frac{5\pi}{4}} \), and \( e^{i\frac{7\pi}{4}} \). -
Equation: \( \frac{x}{4} = -1 \)
Solution: Multiply both sides by 4, \( x = -4 \). -
Equation: \( x - 2 = 2 \)
Solution: Solve for \( x \): \( x = 4 \). -
Equation: \( 16x = 64 \)
Solution: Divide both sides by 16, \( x = 4 \). -
Equation: \( -12x = 48 \)
Solution: Divide both sides by -12, \( x = -4 \).
Now let's summarize the solutions:
- \( x^4 = -1 \) → no real solution (complex solutions)
- \( \frac{x}{4} = -1 \) → \( x = -4 \)
- \( x - 2 = 2 \) → \( x = 4 \)
- \( 16x = 64 \) → \( x = 4 \)
- \( -12x = 48 \) → \( x = -4 \)
From this, we see that:
- The equation \( x^4 = -1 \) has complex solutions (no real solution).
- The equations \( \frac{x}{4} = -1 \) and \( -12x = 48 \) have the solution \( x = -4 \).
- The equations \( x - 2 = 2 \) and \( 16x = 64 \) have the solution \( x = 4 \).
Thus, the equation that has a different solution than the rest is:
\( x^4 = -1 \).