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

Question

UpStudy AI · Accepted Answer

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.
- $$ \sqrt{15} $$ is irrational. - $$ \frac{2.5}{1.3}\sqrt{8} $$ is irrational. - $$ 2.3333333\ldots $$ is rational.

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Arithmetic Questions