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When solving exponential equations, we often come across fractional exponents. A fractional exponent is simply an exponent that is a fraction rather than a whole number. For example, \(a^{1/n}\) represents the n-th root of a. Understanding fractional exponents is essential for rewriting terms in exponential equations.By recognizing that \(\frac{8}{27}\) can be written as \(\left( \frac{2}{3} \right)^3\), we utilize the properties of exponents to simplify the equation. This is pivotal because it allows us to solve the equation more easily. Another key skill is converting between radical forms and fractional exponents, which might come in handy in various contexts.

Another critical step in solving exponential equations is recognizing when the bases are reciprocals. Reciprocal bases multiply to give 1. For example, the reciprocal of \(\frac{3}{2}\) is \(\frac{2}{3}\). This becomes useful because if \(a^x = b^y\) and \(b=\frac{1} {a}\), then \(x = -y\).In our problem, we identified that \(\frac{8}{27}\) can be rewritten as \(\left(\frac{2}{3}\right)^3\). From there, we recognized that \(\frac{2}{3}\) is the reciprocal of \(\frac{3}{2}\), meaning \(\left(\frac{3}{2}\right)^x = \left(\frac{3}{2}\right)^{-3}\). Equating the exponents gives us \(x = -3\). Understanding how reciprocal bases work is fundamental for transforming and solving exponential equations.

After finding a solution to an exponential equation, verifying the solution is crucial. This ensures that our calculations are correct and that the solution truly satisfies the original equation.To verify, substitute the solution back into the original equation. In our case, substituting \(x = -3\) back into \(\left(\frac{3}{2}\right)^x = \frac{8}{27}\), we get \(\left(\frac{3}{2}\right)^{-3} = \frac{8}{27}\). Simplifying the left side, we see both sides are equal, confirming our solution.Verification is an essential step to avoid errors and confirm the accuracy of our solutions. Always take a moment to substitute back and check your work.