Question

consider the statement shown.
the sum of two irrational 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:

Step1: Recall Rational Numbers

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$. Decimals that terminate or repeat are also rational. Irrational numbers cannot be expressed as such (e.g., $\pi$, $\sqrt{2}$).

Step2: Analyze Option A

$A$: $\pi + 4.5$. $\pi$ is irrational, $4.5$ is rational. Their sum is irrational, so this doesn't show the statement is false.

Step3: Analyze Option B

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

Step4: Analyze Option C

$C$: $\sqrt{8}+\sqrt{16}$. $\sqrt{8}=2\sqrt{2}$ (irrational), $\sqrt{16}=4$ (rational). Sum is irrational + rational = irrational. Doesn't show the statement is false.

Step5: Analyze Option D

$D$: $3.45 + 2.81$. Both are terminating decimals (rational). Sum: $3.45+2.81 = 6.26$, rational. Shows statement is false.

Step6: Analyze Option E

$E$: $0 + 3.75$. $0$ and $3.75$ (rational). Sum is $3.75$, rational. Shows statement is false.

Step7: Analyze Option F

$F$: $4.7+\frac{1}{2}$. $4.7$ (terminating decimal, rational) and $\frac{1}{2}$ (rational). Sum: $4.7 + 0.5=5.2$, rational. Shows statement is false.

Answer:

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