Question

is the number \\(\sqrt{32} \cdot 4\\) rational or irrational? \
\boxed{rational} \boxed{irrational}

Explanation:

Step1: Simplify $\sqrt{32}$

We know that $32 = 16\times2$, so $\sqrt{32}=\sqrt{16\times2}$. Using the property of square roots $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ (where $a = 16$ and $b = 2$), we get $\sqrt{16\times2}=\sqrt{16}\cdot\sqrt{2}=4\sqrt{2}$.

Step2: Multiply by 4

Now we multiply $\sqrt{32}$ (which we simplified to $4\sqrt{2}$) by 4. So we have $4\sqrt{2}\times4 = 16\sqrt{2}$.

Step3: Determine if rational or irrational

A rational number can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. An irrational number is a number that cannot be expressed in this form. Since $\sqrt{2}$ is irrational (it cannot be written as a fraction of two integers) and 16 is a non - zero integer, the product $16\sqrt{2}$ is irrational (because the product of a non - zero rational number and an irrational number is irrational).

Answer:

irrational