the sum of two rational numbers is always irrational. choose all of the expressions that show that this…

Explanation:

Step1: Recall rational - number definition

A rational number can be written as $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q\neq0$. Decimals that terminate or repeat are rational.

Step2: Analyze option A

$\pi$ is irrational, $4.5=\frac{9}{2}$ is rational. Their sum $\pi + 4.5$ is irrational, so it doesn't disprove the statement.

Step3: Analyze option B

$\frac{1}{4}+\frac{2}{3}=\frac{3 + 8}{12}=\frac{11}{12}$. Both $\frac{1}{4}$ and $\frac{2}{3}$ are rational, and their sum $\frac{11}{12}$ is rational, disproving the statement.

Step4: Analyze option C

$\sqrt{8}=2\sqrt{2}$ is irrational, $\sqrt{16} = 4$ is rational. Their sum $\sqrt{8}+\sqrt{16}=2\sqrt{2}+4$ is irrational, so it doesn't disprove the statement.

Step5: Analyze option D

$3.45=\frac{345}{100}$ and $2.81=\frac{281}{100}$ are rational. $3.45 + 2.81=6.26=\frac{626}{100}$, sum is rational, disproving the statement.

Step6: Analyze option E

$0$ and $3.75=\frac{15}{4}$ are rational. $0 + 3.75=3.75$, sum is rational, disproving the statement.

Step7: Analyze option F

$4.\overline{7}=4\frac{7}{9}=\frac{43}{9}$ and $\frac{1}{2}$ are rational. $4.\overline{7}+\frac{1}{2}=\frac{43}{9}+\frac{1}{2}=\frac{86 + 9}{18}=\frac{95}{18}$, sum is rational, disproving the statement.

Answer:

B. $\frac{1}{4}+\frac{2}{3}$, D. $3.45 + 2.81$, E. $0+3.75$, F. $4.\overline{7}+\frac{1}{2}$