State whether the following numbers are rational, irrational or neither: \( \begin{array}{llll}\text { (a) } 0,25 & \text { (b) }-4 & \text { (c) } \pi+8 & \text { (d) } \sqrt{11} \\ \text { (e) } \sqrt{-11} & \text { (f) } \sqrt{9} & \text { (g) } \frac{2}{3} & \text { (h) } \frac{5}{0}\end{array} \) \( \begin{array}{llll}\text { (i) } \frac{\pi}{\pi} & \text { (j) } 0,3232323232323232 \ldots & \text { (k) } 0,6820045370518346 \text {. } \\ \text { (l) } \sqrt[3]{-8} & \text { (m) } \sqrt[3]{-7} & \text { (n) } \frac{\pi}{3} & \text { (o) } \sqrt{\frac{4}{9}} \\ \text { (p) } 2 \pi & \text { (q) } 0 \pi & \text { (r) } \sqrt[4]{-16} & \text { (s) } \sqrt{4+9}\end{array} \)

Let's classify each of the given numbers as **rational**, **irrational**, or **neither** based on their properties. ### Definitions: - **Rational Number**: A number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). - **Irrational Number**: A real number that cannot be expressed as a simple fraction; its decimal goes on forever without repeating. - **Neither**: Numbers that are not real, such as those involving imaginary units (\(i\)). ### Classification: **(a) \(0.25\)** - **Rational**: \(0.25 = \frac{1}{4}\). **(b) \(-4\)** - **Rational**: \(-4\) is an integer, which is a subset of rational numbers. **(c) \(\pi + 8\)** - **Irrational**: \(\pi\) is irrational; adding a rational number (8) to an irrational number remains irrational. **(d) \(\sqrt{11}\)** - **Irrational**: \(\sqrt{11}\) cannot be expressed as a simple fraction. **(e) \(\sqrt{-11}\)** - **Neither**: This is an imaginary number since it involves the square root of a negative number. **(f) \(\sqrt{9}\)** - **Rational**: \(\sqrt{9} = 3\), which is an integer. **(g) \(\frac{2}{3}\)** - **Rational**: Explicitly given as a fraction of integers. **(h) \(\frac{5}{0}\)** - **Neither**: Division by zero is undefined. **(i) \(\frac{\pi}{\pi}\)** - **Rational**: Assuming \(\pi \neq 0\), \(\frac{\pi}{\pi} = 1\). **(j) \(0.3232323232\ldots\)** - **Rational**: This is a repeating decimal which can be expressed as \(\frac{32}{99}\). **(k) \(0.6820045370518346\ldots\)** - **Irrational**: The decimal does not terminate or repeat, suggesting it cannot be expressed as a fraction. **(l) \(\sqrt[3]{-8}\)** - **Rational**: \(\sqrt[3]{-8} = -2\). **(m) \(\sqrt[3]{-7}\)** - **Irrational**: \(\sqrt[3]{-7}\) cannot be expressed as a simple fraction. **(n) \(\frac{\pi}{3}\)** - **Irrational**: Dividing an irrational number (\(\pi\)) by a rational number (3) remains irrational. **(o) \(\sqrt{\frac{4}{9}}\)** - **Rational**: \(\sqrt{\frac{4}{9}} = \frac{2}{3}\). **(p) \(2\pi\)** - **Irrational**: Multiplying an irrational number by a rational number (2) remains irrational. **(q) \(0 \cdot \pi\)** - **Rational**: \(0 \cdot \pi = 0\), which is rational. **(r) \(\sqrt[4]{-16}\)** - **Neither**: This is a non-real number since it involves an even root of a negative number. **(s) \(\sqrt{4 + 9}\)** - **Irrational**: \(\sqrt{4 + 9} = \sqrt{13}\), which cannot be expressed as a simple fraction. ### Summary: | Item | Classification | |------|----------------| | **(a)** | Rational | | **(b)** | Rational | | **(c)** | Irrational | | **(d)** | Irrational | | **(e)** | Neither | | **(f)** | Rational | | **(g)** | Rational | | **(h)** | Neither | | **(i)** | Rational | | **(j)** | Rational | | **(k)** | Irrational | | **(l)** | Rational | | **(m)** | Irrational | | **(n)** | Irrational | | **(o)** | Rational | | **(p)** | Irrational | | **(q)** | Rational | | **(r)** | Neither | | **(s)** | Irrational |