Question

select the expression that results in a rational number.
a. $sqrt3{36} + sqrt{64}$
b. $-0.137137 + sqrt3{9}$
c. $sqrt{80} + (-0.overline{6})$
d. $0.overline{1} + sqrt{169}$

Explanation:

Step1: Evaluate Option A

$\sqrt[3]{36} + \sqrt{64} = \sqrt[3]{36} + 8$. $\sqrt[3]{36}$ is irrational, so sum is irrational.

Step2: Evaluate Option B

$-0.137137 + \sqrt[3]{9}$. $\sqrt[3]{9}$ is irrational, so sum is irrational.

Step3: Evaluate Option C

$\sqrt{80} + (-0.\overline{6}) = 4\sqrt{5} - \frac{2}{3}$. $\sqrt{5}$ is irrational, so sum is irrational.

Step4: Evaluate Option D

$0.\overline{1} + \sqrt{169} = \frac{1}{9} + 13 = \frac{1 + 117}{9} = \frac{118}{9}$. This is rational.

Answer:

D. $0.\overline{1} + \sqrt{169}$