3.3 Simplify the following: \( \frac{\sqrt{9}}{\sqrt{16}}+\frac{\sqrt[3]{8}}{\sqrt[3]{27}} \)

To simplify the expression \( \frac{\sqrt{9}}{\sqrt{16}}+\frac{\sqrt[3]{8}}{\sqrt[3]{27}} \), we can evaluate each term separately. First, we simplify \( \frac{\sqrt{9}}{\sqrt{16}} \): \[ \sqrt{9} = 3 \quad \text{and} \quad \sqrt{16} = 4 \] So, \[ \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \] Next, we simplify \( \frac{\sqrt[3]{8}}{\sqrt[3]{27}} \): \[ \sqrt[3]{8} = 2 \quad \text{and} \quad \sqrt[3]{27} = 3 \] So, \[ \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \] Now we combine both simplified fractions: \[ \frac{3}{4} + \frac{2}{3} \] To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12. Converting both fractions: \[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \] \[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \] Now, we can add: \[ \frac{9}{12} + \frac{8}{12} = \frac{17}{12} \] Thus, the final simplified result is: \[ \frac{17}{12} \]