Algebra
# Exponential Functions

\[\large 2^x=\dfrac{1}{2^y}, \quad\quad\quad x + y = \ ? \]

\[\large 2^{x} = 3^{y} = 12^{z} \]

If the equation above is fulfilled for non-zero values of \(x,y,z,\) find the value of \(\frac { z(x+2y) }{ xy }\).

\[\large{ \begin{cases}

ab = a^b \\\\
\frac{a}{b} = a ^ {3b} \\ \end{cases}} \]

\( a\) and \(b\) are real numbers such that \( a > 1 \) and \( b \neq 0 \), satisfying the above system.

Find \( b ^ { -a}. \)

\[\large 5^{2x}+5^5=5^{x+3}+5^{x+2}\]

Find the sum of the solutions of \(x\) that satisfy the equation above.