Question: Solve the following elementary exponential equation: \[ \left( \frac{1}{3} \right)^{4x+2} =…

Solve the following elementary exponential equation:

\[ \left( \frac{1}{3} \right)^{4x+2} = 27^x \]

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\[ x = \boxed{\phantom{0}} \]

Solution

To solve the given exponential equation: \[ \left(\frac{1}{3}\right)^{4x+2} = 27^x \] We can start by rewriting both sides in terms of the same base. Rewrite \(\frac{1}{3}\) as \(3^{-1}\) and \(27\) as \(3^3\): \[ \left(3^{-1}\right)^{4x+2} = (3^3)^x \] Simplify the exponents: \[ 3^{-(4x+2)} = 3^{3x} \] Since the bases are the same, we can set the exponents equal to each other: \[ -(4x+2) = 3x \] Solve for \(x\): First, distribute the negative sign: \[ -4x - 2 = 3x \] Add \(4x\) to both sides: \[ -2 = 7x \] Divide both sides by 7: \[ x = -\frac{2}{7} \] Thus, the solution is: \(x = -\frac{2}{7}\)