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

To simplify the expression: \[ 9^{-1} + 4^{-1} \] let's break it down step by step. ### Step 1: Understand Negative Exponents A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Therefore: \[ 9^{-1} = \frac{1}{9} \] \[ 4^{-1} = \frac{1}{4} \] ### Step 2: Rewrite the Expression with Fractions Substitute the negative exponents with their fractional equivalents: \[ 9^{-1} + 4^{-1} = \frac{1}{9} + \frac{1}{4} \] ### Step 3: Find a Common Denominator To add the two fractions, they must have the same denominator. The least common denominator (LCD) of 9 and 4 is 36. \[ \frac{1}{9} = \frac{4}{36} \quad \text{(multiplied numerator and denominator by 4)} \] \[ \frac{1}{4} = \frac{9}{36} \quad \text{(multiplied numerator and denominator by 9)} \] ### Step 4: Add the Fractions Now that both fractions have the same denominator, add them together: \[ \frac{4}{36} + \frac{9}{36} = \frac{4 + 9}{36} = \frac{13}{36} \] ### Final Simplified Expression \[ 9^{-1} + 4^{-1} = \frac{13}{36} \] **Answer:** \(\frac{13}{36}\)