1. If \( a=b^{c}, b=c^{a} \), and \( c=a^{b} \), then \( a b c=1 \) (True/False)

To determine if the statement \( a=b^{c}, b=c^{a}, c=a^{b} \) implies that \( abc=1 \), we need to analyze the given equations and see if they lead to the conclusion that the product of \( a \), \( b \), and \( c \) is equal to 1. Given: 1. \( a=b^{c} \) 2. \( b=c^{a} \) 3. \( c=a^{b} \) We can start by substituting the values of \( a \), \( b \), and \( c \) from the given equations into the equation \( abc=1 \) to see if it holds true. Let's substitute the values of \( a \), \( b \), and \( c \) from the given equations into the equation \( abc=1 \) and simplify to determine if the statement is true or false. Simplify the expression by following steps: - step0: Solution: \(abc\) The expression \( b^{c}c^{a}a^{b} \) cannot be simplified further. Therefore, the statement \( a=b^{c}, b=c^{a}, c=a^{b} \) does not imply that \( abc=1 \). The statement is false.