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The question from the worksheet asks about the rationality of the numbers \( 3\sqrt{7} \) and \( 5\sqrt{7} \) and further explores their properties when multiplied, divided, and added. Let's break it down:

1. **Are \( 3\sqrt{7} \) and \( 5\sqrt{7} \) rational or irrational?**
   - A rational number is a number that can be expressed as the ratio of two integers (like \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \)).
   - The square root of a non-perfect square (such as \( \sqrt{7} \)) is irrational. Therefore, both \( 3\sqrt{7} \) and \( 5\sqrt{7} \) involve irrational components.
   - Since multiplying a rational number (like 3 or 5) with an irrational number (like \( \sqrt{7} \)) results in an irrational number, both \( 3\sqrt{7} \) and \( 5\sqrt{7} \) are irrational numbers.

2. **Is their product rational?**
   - The product is \( (3\sqrt{7})(5\sqrt{7}) = 15 \times \sqrt{7} \times \sqrt{7} = 15 \times 7 = 105 \), which is a rational number.
   
3. **Is their quotient rational?**
   - The quotient is \( \frac{3\sqrt{7}}{5\sqrt{7}} = \frac{3}{5} \), which is a rational number because both 3 and 5 are integers, and \( \sqrt{7} \) cancels out.

4. **Is their sum rational?**
   - The sum is \( 3\sqrt{7} + 5\sqrt{7} = (3+5)\sqrt{7} = 8\sqrt{7} \).
   - Since \( \sqrt{7} \) is irrational, multiplying it by 8 still results in an irrational number, so the sum is irrational.

### Summary:
- \( 3\sqrt{7} \) and \( 5\sqrt{7} \) are irrational.
- Their product is rational.
- Their quotient is rational.
- Their sum is irrational.

Would you like further details or explanations? Here are some related questions to deepen your understanding:

1. What are the conditions for a number to be rational?
2. Can the product of two irrational numbers ever be rational?
3. How does the nature of square roots of non-perfect squares affect rationality?
4. How do we classify numbers like \( \sqrt{2} \) or \( \sqrt{3} \)?
5. Can you find examples where the sum of two irrational numbers is rational?

**Tip:** Always remember, the sum of two irrational numbers can still be irrational, depending on how they interact!

Apakah bilangan 3√7 dan 5√7 bersifat rasional atau tidak? Jelaskan! Apakah hasil kalinya, baginya, dan jumlahnya merupakan bilangan rasional?

This question explores whether the numbers 3√7 and 5√7 are rational or irrational, and whether their product, quotient, and sum are rational. It covers concepts like the properties of square roots, and rational vs. irrational numbers. Suitable for Grades 7-9.

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