Question

classify the numbers as rational or irrational. 7, √18, 7·√18, 7 + √18. rational, irrational

Explanation:

Step1: Recall definitions

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. An irrational number is a non - repeating, non - terminating decimal and cannot be expressed as a fraction of two integers. Also, the product or sum of a non - zero rational number and an irrational number is irrational, and the sum of a rational number and an irrational number is irrational.

Step2: Classify 7

The number 7 can be written as $\frac{7}{1}$, where $p = 7$ and $q=1$ are integers and $q
eq0$. So, 7 is a rational number.

Step3: Classify $\sqrt{18}$

Simplify $\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}$. Since $\sqrt{2}$ is an irrational number (it is a non - repeating, non - terminating decimal), and $3\sqrt{2}$ is a non - zero multiple of an irrational number, $\sqrt{18}$ is an irrational number.

Step4: Classify $7\cdot\sqrt{18}$

We know that 7 is a rational number (non - zero) and $\sqrt{18}$ is an irrational number. The product of a non - zero rational number and an irrational number is irrational. So, $7\cdot\sqrt{18}$ is an irrational number.

Step5: Classify $7+\sqrt{18}$

We know that 7 is a rational number and $\sqrt{18}$ is an irrational number. The sum of a rational number and an irrational number is irrational. So, $7 + \sqrt{18}$ is an irrational number.

Answer:

rational: 7
irrational: $\sqrt{18}$, $7\cdot\sqrt{18}$, $7+\sqrt{18}$