dicate whether each of the following numbers is rational or irrational: 1.1 \( \sqrt{15} \) \( 1.2 \frac{2,5}{1.3} \sqrt[2]{8} \) \( 1.42,3333333 \ldots \)

1. Consider the number \( \sqrt{15} \). Since \(15\) is not a perfect square, \(\sqrt{15}\) cannot be expressed as a ratio of two integers. Thus, \(\sqrt{15}\) is irrational. 2. Consider the number \[ \frac{2.5}{1.3}\sqrt{8}. \] First, express the decimal numbers as fractions: \[ 2.5 = \frac{5}{2} \quad \text{and} \quad 1.3 = \frac{13}{10}. \] Compute the ratio: \[ \frac{2.5}{1.3} = \frac{\frac{5}{2}}{\frac{13}{10}} = \frac{5}{2} \times \frac{10}{13} = \frac{50}{26} = \frac{25}{13}. \] Also, simplify \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}. \] Now, the number becomes: \[ \frac{25}{13} \cdot 2\sqrt{2} = \frac{50}{13}\sqrt{2}. \] Since \(\sqrt{2}\) is irrational and the product of a nonzero rational number and an irrational number is irrational, \(\frac{50}{13}\sqrt{2}\) is irrational. 3. Consider the number \[ 2.3333333\ldots \] Notice that the digit \(3\) repeats indefinitely. This repeating decimal can be written as: \[ 2.3333333\ldots = 2 + 0.3333333\ldots = 2 + \frac{1}{3} = \frac{7}{3}. \] Since \(\frac{7}{3}\) is a fraction of integers, the number is rational. Thus, the answers are: - \( \sqrt{15} \) is irrational. - \( \frac{2.5}{1.3}\sqrt{8} \) is irrational. - \( 2.3333333\ldots \) is rational.