Question

the number w is irrational. which statement about $w \cdot \sqrt{6}$ is true? $w \cdot \sqrt{6}$ is rational. $w \cdot \sqrt{6}$ is irrational. $w \cdot \sqrt{6}$ can be rational or irrational, depending on the value of w.

Explanation:

Step1: Recall properties of irrational numbers

An irrational number is a number that cannot be expressed as a fraction of two integers. $\sqrt{6}$ is irrational (since 6 is not a perfect square, so its square root is non - repeating and non - terminating in decimal form).

Step2: Analyze the product of irrational numbers

Case 1: Let's take an example. Suppose $w = \sqrt{6}$. Then $w\cdot\sqrt{6}=\sqrt{6}\cdot\sqrt{6} = 6$, which is rational.
Case 2: Now, suppose $w=\sqrt{2}$ (which is irrational). Then $w\cdot\sqrt{6}=\sqrt{2}\cdot\sqrt{6}=\sqrt{12} = 2\sqrt{3}$, which is irrational.

So, the product $w\cdot\sqrt{6}$ can be rational or irrational depending on the value of $w$.

Answer:

$w\cdot\sqrt{6}$ can be rational or irrational, depending on the value of $w$.