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Understanding how to solve equations is crucial in algebra. This involves finding the value of the variable that makes the equation true. The given exercise asks us to solve the equation: \[ \bigg(\frac{4}{3}\bigg)^x = \frac{27}{64} \] The key to solving this type of equation is to express both sides with the same base. Once we do that, it becomes much simpler to solve for the variable by setting the exponents equal. This is because exponential functions with the same base are equal if and only if their exponents are equal.

By identifying common bases, we can simplify exponential equations. In the given exercise, we need to express \( \frac{27}{64} \) in terms of a common base with \( \bigg(\frac{4}{3}\bigg) \). We rewrite \( \frac{27}{64} \) as: \[ \frac{27}{64} = \frac{3^3}{4^3} = \big(\frac{3}{4}\big)^3 \] This converts the original equation to: \[ \bigg(\frac{4}{3}\bigg)^x = \big(\frac{3}{4}\big)^3 \] Converting \( \big(\frac{3}{4}\big)^3 \) to a common base, we get: \[ \bigg(\frac{3}{4}\bigg)^3 = \bigg(\big(\frac{4}{3}\big)^{-1}\bigg)^3 = \bigg(\frac{4}{3}\bigg)^{-3} \] The equation now looks like: \[ \bigg(\frac{4}{3}\bigg)^x = \big(\frac{4}{3}\big)^{-3} \] With both sides having the same base, we can easily move to the next step.

Now that we have the equation: \[ \bigg(\frac{4}{3}\bigg)^x = \big(\frac{4}{3}\big)^{-3} \] We can solve for \( x \) by equating the exponents since the bases are identical. Thus: \[ x = -3 \] Simplifying exponents and equating them when the bases are the same is a method used to solve exponential equations. This technique helps us find the variable's value quickly and efficiently. It’s a valuable skill in algebra, letting us tackle a variety of exponential problems. Practice recognizing common bases and matching exponents to solve similar equations.