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The question is asking which sum or product of the given options results in a rational number. Let's go through each choice step by step:

**Option A: \(\sqrt{15} + \sqrt{\frac{1}{9}}\)**
- \(\sqrt{15}\) is irrational because 15 is not a perfect square.
- \(\sqrt{\frac{1}{9}} = \frac{1}{3}\) is rational.
- Adding an irrational number \(\sqrt{15}\) to a rational number \(\frac{1}{3}\) will give an irrational result.
- **This is not rational.**

**Option B: \(\sqrt{16} + \sqrt{10}\)**
- \(\sqrt{16} = 4\) is rational.
- \(\sqrt{10}\) is irrational because 10 is not a perfect square.
- Adding a rational number (4) to an irrational number (\(\sqrt{10}\)) results in an irrational number.
- **This is not rational.**

**Option C: \((\sqrt{25})^3 \times (\sqrt{14})^2\)**
- \(\sqrt{25} = 5\), so \((\sqrt{25})^3 = 5^3 = 125\) (rational).
- \(\sqrt{14}\) is irrational, but squaring it gives \((\sqrt{14})^2 = 14\) (rational).
- The product of two rational numbers (125 and 14) is rational.
- **This is rational.**

**Option D: \((\sqrt{36})^2 \times (\sqrt{13})^3\)**
- \(\sqrt{36} = 6\), so \((\sqrt{36})^2 = 6^2 = 36\) (rational).
- \(\sqrt{13}\) is irrational, so \((\sqrt{13})^3\) remains irrational.
- The product of a rational number (36) and an irrational number is irrational.
- **This is not rational.**

Thus, the correct answer is **C: \((\sqrt{25})^3 \times (\sqrt{14})^2\)**.

Would you like further details or have any additional questions?

Here are 5 follow-up questions you might consider:
1. What makes a number rational or irrational?
2. How can you tell if the square root of a number is rational?
3. What happens when you multiply a rational number with an irrational number?
4. Can the sum of two irrational numbers be rational? Why or why not?
5. What are some examples of irrational numbers and their properties?

**Tip**: When determining if a number is rational, always check if it can be expressed as a fraction of two integers.

Which sum or product is rational? A) √15 + √(1/9), B) √16 + √10, C) (√25)^3(√14)^2, D) (√36)^2(√13)^3

Explore how to determine whether a sum or product involving square roots is rational or irrational. This problem covers key concepts such as square roots, exponents, and the difference between rational and irrational numbers, suitable for Grades 9-11.

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