Chapter 6.6, Problem 4WE

a.

To determine

Find whether the number $\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$ is rational or irrational.

Answer to Problem 4WE

The number $\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$ is irrational.

Explanation of Solution

Given information: The given number is $\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$ .

Calculation: The given number $\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$ is irrational because the number $\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$ is a non-repeating and non-terminated number when written in its simplified form, which cannot be expressed as the ratio of two integers in the form $\frac{p}{q}$ where $q\ne 0$ . Hence the number $\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$ is irrational.

b.

To determine

Find whether the number $\sqrt{\frac{1}{2}}\cdot \sqrt{\frac{1}{8}}$ is rational or irrational.

Answer to Problem 4WE

The number $\sqrt{\frac{1}{2}}\cdot \sqrt{\frac{1}{8}}$ is rational.

Explanation of Solution

Given information: The given number is $\sqrt{\frac{1}{2}}\cdot \sqrt{\frac{1}{8}}$ .

Calculation: The given number $\sqrt{\frac{1}{2}}\cdot \sqrt{\frac{1}{8}}$ is rational because the number $\sqrt{\frac{1}{2}}\cdot \sqrt{\frac{1}{8}}$ is a terminated decimal number when written its simplified form which can be expressed as the ratio of two integers in the form $\frac{p}{q}$ where $q\ne 0$ . Hence the number $\sqrt{\frac{1}{2}}\cdot \sqrt{\frac{1}{8}}$ is rational.