14 Is each of these numbers rational or irrational? Give a reason for each answer. \( \begin{array}{llll}\text { a } \sqrt{3}+6 & \text { b } \sqrt{3+6} & \text { c } \sqrt{64}+\sqrt[3]{64} & \text { d } \sqrt[3]{8}+\sqrt[3]{19}\end{array} \)

Let's determine whether each of the given numbers is **rational** or **irrational** and provide reasons for each answer. ### a) \( \sqrt{3} + 6 \) **Answer:** **Irrational** **Reason:** - \( \sqrt{3} \) is an **irrational** number because 3 is not a perfect square. - 6 is a **rational** number. - The sum of a rational number and an irrational number is **irrational**. ### b) \( \sqrt{3 + 6} \) **Answer:** **Rational** **Reason:** - Simplify the expression inside the square root: \( 3 + 6 = 9 \). - \( \sqrt{9} = 3 \), which is a **rational** number. ### c) \( \sqrt{64} + \sqrt[3]{64} \) **Answer:** **Rational** **Reason:** - \( \sqrt{64} = 8 \) because \( 8^2 = 64 \). - \( \sqrt[3]{64} = 4 \) because \( 4^3 = 64 \). - Both 8 and 4 are **rational** numbers. - The sum of two rational numbers is **rational**: \( 8 + 4 = 12 \). ### d) \( \sqrt[3]{8} + \sqrt[3]{19} \) **Answer:** **Irrational** **Reason:** - \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \) (which is **rational**). - \( \sqrt[3]{19} \) is **irrational** because 19 is not a perfect cube. - The sum of a rational number and an irrational number is **irrational**: \( 2 + \sqrt[3]{19} \). --- **Summary:** - **a)** Irrational - **b)** Rational - **c)** Rational - **d)** Irrational