Question

consider the statement: \the sum of a rational number and an irrational num choose all of the expressions that show that this statement is false. a. $sqrt{25}+pi$ b. $0.56+pi$ c. $\frac{7}{8}+sqrt{13}$ d. $pi+sqrt{17}$ e. $0.45 + 0.96$ f. $sqrt{18}+sqrt{21}$

Explanation:

Step1: Identify the target statement

The full implied statement to disprove is: "The sum of a rational number and an irrational number is rational" (since we need expressions that show this is false, i.e., sums that are irrational, matching the rational+irrational form).

Step2: Classify each option's terms

Option A:

$\sqrt{25}=5$ (rational), $\pi$ (irrational). Sum: $5+\pi$ (irrational). This fits rational+irrational=irrational, disproving the statement.

Option B:

$0.56$ (rational), $\pi$ (irrational). Sum: $0.56+\pi$ (irrational). Fits the required form.

Option C:

$\frac{7}{8}$ (rational), $\sqrt{13}$ (irrational). Sum: $\frac{7}{8}+\sqrt{13}$ (irrational). Fits the required form.

Option D:

$\pi$ (irrational), $\sqrt{17}$ (irrational). Sum of two irrationals: does not test rational+irrational, so irrelevant.

Option E:

$0.45$ (rational), $0.96$ (rational). Sum is rational: does not disprove the statement.

Option F:

$\sqrt{18}$ (irrational), $\sqrt{21}$ (irrational). Sum of two irrationals: irrelevant to the statement.

Answer:

A. $\sqrt{25} + \pi$
B. $0.56 + \pi$
C. $\frac{7}{8} + \sqrt{13}$