Question

the sum of two rational numbers is always irrational.
choose all of the expressions that show that this statement is false.
a. $\pi + 4.5$
b. $\frac{1}{4}+\frac{2}{3}$
c. $\sqrt{8}+\sqrt{16}$
d. $3.45 + 2.81$
e. $0 + 3.75$
f. $4.7+\frac{1}{2}$

Explanation:

for each option:

Option A:

Step1: Identify number types

$\pi$ is irrational, $4.5$ is rational.

Step2: Analyze sum type

Sum of irrational and rational is irrational. This doesn't show the statement is false.

Option B:

Step1: Identify number types

$\frac{1}{4}$ and $\frac{2}{3}$ are rational (fractions of integers).

Step2: Calculate sum

$\frac{1}{4}+\frac{2}{3}=\frac{3 + 8}{12}=\frac{11}{12}$, which is rational. So this shows the statement is false.

Option C:

Step1: Identify number types

$\sqrt{8}=2\sqrt{2}$ (irrational), $\sqrt{16}=4$ (rational).

Step2: Analyze sum type

Sum of irrational and rational is irrational. Doesn't show the statement is false.

Option D:

Step1: Identify number types

$3.45$ and $2.81$ are rational (terminating decimals).

Step2: Calculate sum

$3.45 + 2.81=6.26$, which is rational. Shows the statement is false.

Option E:

Step1: Identify number types

$0$ and $3.75$ are rational.

Step2: Calculate sum

$0 + 3.75 = 3.75$, which is rational. Shows the statement is false.

Option F:

Step1: Identify number types

$4.7$ (rational, terminating decimal) and $\frac{1}{2}=0.5$ (rational).

Step2: Calculate sum

$4.7+0.5 = 5.2$, which is rational. Shows the statement is false.

Answer:

B. $\boldsymbol{\frac{1}{4}+\frac{2}{3}}$, D. $\boldsymbol{3.45 + 2.81}$, E. $\boldsymbol{0 + 3.75}$, F. $\boldsymbol{4.7+\frac{1}{2}}$