Simplify the following expression. \[ 9^{-1}+4^{-1} \]

To simplify the expression \( 9^{-1} + 4^{-1} \), we start by rewriting the terms using their reciprocals: \[ 9^{-1} = \frac{1}{9} \quad \text{and} \quad 4^{-1} = \frac{1}{4} \] Now, substituting these into the expression gives us: \[ \frac{1}{9} + \frac{1}{4} \] To add these fractions, we need a common denominator. The least common multiple of 9 and 4 is 36. We’ll convert each fraction: \[ \frac{1}{9} = \frac{4}{36} \quad \text{(since \(1 \times 4 = 4\) and \(9 \times 4 = 36\))} \] \[ \frac{1}{4} = \frac{9}{36} \quad \text{(since \(1 \times 9 = 9\) and \(4 \times 9 = 36\))} \] Now we can add the two fractions: \[ \frac{4}{36} + \frac{9}{36} = \frac{4 + 9}{36} = \frac{13}{36} \] Thus, the simplified expression is: \[ \boxed{\frac{13}{36}} \]