Chapter 8.7, Problem 17WE

To determine

To show: that the $\sqrt[4]{8}$ is irrational number.

Explanation of Solution

Given information:

The number is $\sqrt[4]{8}$ .

Proof: since, $\sqrt[4]{8}$ is a root of the equation $x^4-8=0$ . The only possible rational roots of this equation are $\pm 1,\pm 2,\pm 4\text{ and }\pm 8$ .

None of these eight numbers satisfies the equation.

Since, equation $x^4-8=0$ has no rational roots,

Thus, $\sqrt[4]{8}$ is irrational.