The given equationThe equation \ is:

(a^((a^c)^c = a\) can be simplified and analyzedc)^c = a

We aim to analyze as follows:

Step 1 and simplify this equation: Simplify the expression

Using the property of exponents. To do so, let's break it down step by step.

Step $(x^m)^n = x^{m \cdot n}$, we rewrite1: Simplify the left-hand side

Using the rule of exponents (x^ \((a^c)^c as:

\[ (a} = a^{c^2}.$$ Thus, the equation becomes^c)^c = a^{c \cdot c} = a^{c^2} \: $$a^{c^2}] Thus, the equation becomes: \[ a^{c^2} = a = a.$$ ### Step 2: Eliminate the base $$a$$ \] ### Step 2: Use the property of equality of exponents For basesAssuming $$a \neq 0$$ and \(a \neq 1\ that are the same, \(x^m = x^n \implies m = n), we can equate the exponents: $$c^2 = 1.$$ \) (assuming $$x \neq 0\### Step 3: Solve for \(c$$ The solutions to $$c^2 = 1$$ and $$x \neq 1$$). Here,) are: $$c = the bases are the same (\(a1 \quad \text{or} \quad c = -1.$$ ### Solution\)), so we equate the exponents: $$c^2 = 1 \: For the given equation $$(a^c)^c = a$$, the solutions for $$c$$ are: \[ c] ### Step 3: Solve for $$c$$ The equation $$c^2 = 1$$ has two solutions: = 1 \quad \text{or} \quad c = -1.$$ ### Special Cases: 1. If$$c = 1 \quad \text{or} \quad c = -1$$ ### Step 4: Verify the solutions 1. If $$c = 1$$, then $$(a^c)^c = (a^1)^1 = a \(a = 1$$, the equation holds for all $$c$$, since $$1^c =$$, which is true. 2. If $$c =1$$. 2. If $$-1$$, then $$(aa = 0$$, the equation becomes invalid unless $$^c)^c = (a^{-1})^{-1} = a$$, which is also true. c > 0\), as \(0^c = 0### Final Answer: The solutions are: \\) for positive $$c$$. Let me know if you want more details or have any questions! ### Related Questions: 1. What are the properties of exponents used in solving this equation? 2. How does the solution change if $$a = 1$$ or[ c = 1 \quad \text{or} \quad c = -1 \] Would you like further details or additional examples related to this topic? --- **Relative Questions:** 1. How $$a = 0$$? 3. What happens if $$c$$ is a complex number instead of a real number? 4. Can this equation have does this property change if $$a = 1$$ or $$a = 0\ more solutions if the base \(a$$ is allowed to vary)? 2. Can you generalize this equation for different forms of? 5. How can this equation be applied in logarithmic or exponents? 3. What happens if we introduce complex numbers for exponential functions? ### Tip: Always check for special cases like $$\(c$$? 4. How would this equation be solved grapha = 1\) or \(a = 0\ically? 5. What if $$a$$ is a negative) when solving equations involving exponents, as they may yield exceptions number? Does it impact the solutions? **Tip:** Always remember the rules of exponents when simplifying or solving such equations; they are key