Exponential Functions
If \( 3^n = 5 \) and \( 4^m = 8 \), what is the value of \[\large 9^{n+m} ? \]
Let \[ f(x) = \left( \dfrac12 \right)^{x-5} - 64, \] and suppose that the graphs of \( y = |f(x)| \) and \( y=k \) meet in the first quadrant.
Find the number of all the positive integers \( k \) that satisfy the above condition.
For all positive integers \(n\), let \(\alpha_n\) and \(\beta_n\) be the roots of the quadratic equation \[nx^2-x+n(n+1)=0.\] If \[\left(\frac{1}{\alpha_1}+\frac{1}{\beta_1}\right)+\left(\frac{1}{\alpha_2}+\frac{1}{\beta_2}\right)+ \cdots +\left(\frac{1}{\alpha_{7}}+\frac{1}{\beta_{7}}\right)=\frac{p}{q},\] where \(p\) and \(q\) are coprime positive integers, what is \(p+q\)?
What is the sum of all possible integer values of \(x\) such that
\[ \large (x^2-55x+755)^{(x^2-62x+936 )} =1 ? \]
Solve for \(x \):
\[ 3 = 4^x + 5^x + 6^x \]