https://file.aiquickdraw.com/mathbot/file/1726008256501n8smkmp7.jpg

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!

Select the expression that results in a rational number.

Analyze different expressions containing square and cube roots to identify which results in a rational number. This problem reviews rational numbers, repeating decimals, and properties of radicals, ideal for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation