Let's solve each exponential equation and match them to the correct answers.
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Equation: \( 3^{x - 4 + 2x} = 3^{3(2x + 7)} \)
We can simplify the left side: \[ 3^{3x - 4} = 3^{3(2x + 7)} \] The bases are the same, so we can set the exponents equal: \[ 3x - 4 = 6x + 21 \] Now, solve for \( x \): \[ 3x - 6x = 21 + 4 \ -3x = 25 \ x = -\frac{25}{3} \]
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Equation: \( 5^{27} = 5^{-9x} \)
Again, we can equate the exponents: \[ 27 = -9x \] Solve for \( x \): \[ x = -\frac{27}{9} \ x = -3 \]
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Equation: \( 12^{x - 5} = 12^{3x} \)
Set the exponents equal: \[ x - 5 = 3x \] Solve for \( x \): \[ x - 3x = 5 \ -2x = 5 \ x = -\frac{5}{2} \]
Summary of Solutions:
- \( 3^{x - 4 + 2x} = 3^{3(2x + 7)} \) leads to \( x = -\frac{25}{3} \)
- \( 5^{27} = 5^{-9x} \) leads to \( x = -3 \)
- \( 12^{x - 5} = 12^{3x} \) leads to \( x = -\frac{5}{2} \)
Matching Answers:
- The solution \( x = -\frac{25}{3} \) corresponds to \( 3^{x - 4 + 2x} = 3^{3(2x + 7)} \)
- The solution \( x = -3 \) corresponds to \( 5^{27} = 5^{-9x} \)
- The solution \( x = -\frac{5}{2} \) corresponds to \( 12^{x - 5} = 12^{3x} \)
So the correct matches are:
- For \( 3^{x - 4 + 2x} = 3^{3(2x + 7)} \): \( x = -\frac{25}{3} \)
- For \( 5^{27} = 5^{-9x} \): \( x = -3 \)
- For \( 12^{x - 5} = 12^{3x} \): \( x = -\frac{5}{2} \)
1 answer