Let's break down each option to determine which expression results in a rational number:

A. $-0.137137 + \sqrt[3]{9}$

• $-0.137137$ is a repeating decimal that can be expressed as a rational number.
• $\sqrt[3]{9} \approx 2.08008$ is not a rational number (because 9 is not a perfect cube).

Therefore, A does not result in a rational number.

B. $\sqrt{80} + (-0.6)$

• $\sqrt{80} = 4\sqrt{5} \approx 8.944$, which is irrational.
• $-0.6$ is rational, but since the square root of 80 is irrational, the entire expression is irrational.

C. $\sqrt[3]{36} + \sqrt{64}$

• $\sqrt[3]{36} \approx 3.301$, which is irrational.
• $\sqrt{64} = 8$, which is rational, but because $\sqrt[3]{36}$ is irrational, the whole expression is irrational.

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

• $0.\overline{1}$ is a repeating decimal that can be expressed as $\frac{1}{9}$, which is rational.
• $\sqrt{169} = 13$, which is rational.

Since both terms are rational, the entire expression is rational.

Thus, D is the correct answer. It results in a rational number.

Would you like further details or explanations on any step?

Here are 5 related questions to explore:

1. What is the difference between rational and irrational numbers?
2. How do cube roots of non-perfect cubes behave?
3. Why is a repeating decimal considered a rational number?
4. How do you determine if a square root will be rational?
5. What are some common methods for simplifying radical expressions?

Tip: Repeating decimals can always be expressed as fractions, making them rational numbers!