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. $4.\overline{7} + \frac{1}{2}$

Explanation:

To determine which expressions show the statement "The sum of two rational numbers is always irrational" is false, we need to identify expressions where both addends are rational and their sum is rational. A rational number is a number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q
eq0\), or as a terminating or repeating decimal.

Step 1: Analyze Option A

\(\pi\) is an irrational number (it cannot be expressed as a fraction of two integers and has a non - repeating, non - terminating decimal expansion), and \(4.5=\frac{9}{2}\) is rational. Since one of the numbers is irrational, this expression does not help in showing the given statement is false.

Step 2: Analyze Option B

\(\frac{1}{4}\) and \(\frac{2}{3}\) are both rational numbers (they are fractions of integers). Let's calculate their sum: \(\frac{1}{4}+\frac{2}{3}=\frac{3 + 8}{12}=\frac{11}{12}\), which is also a rational number. So this expression shows the statement is false.

Step 3: Analyze Option C

\(\sqrt{8} = 2\sqrt{2}\), and \(\sqrt{2}\) is irrational, so \(\sqrt{8}\) is irrational. \(\sqrt{16}=4\) is rational. Since one number is irrational, this expression does not help in showing the statement is false.

Step 4: Analyze Option D

\(3.45=\frac{345}{100}\) and \(2.81=\frac{281}{100}\) are both rational numbers (terminating decimals). Their sum is \(3.45 + 2.81=6.26=\frac{626}{100}=\frac{313}{50}\), which is rational. So this expression shows the statement is false.

Step 5: Analyze Option E

\(0\) (which can be written as \(\frac{0}{1}\)) and \(3.75=\frac{15}{4}\) are both rational numbers. Their sum is \(0 + 3.75 = 3.75=\frac{15}{4}\), which is rational. So this expression shows the statement is false.

Step 6: Analyze Option F

\(4.\overline{7}\) is a repeating decimal (rational, since it can be expressed as a fraction) and \(\frac{1}{2}\) is rational. Let's convert \(4.\overline{7}\) to a fraction. Let \(x = 4.\overline{7}\), then \(10x=47.\overline{7}\), and \(10x - x=47.\overline{7}-4.\overline{7}\), \(9x = 43\), so \(x=\frac{43}{9}\). Then \(\frac{43}{9}+\frac{1}{2}=\frac{86 + 9}{18}=\frac{95}{18}\), which is rational. So this expression shows the statement is false.

Answer:

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