https://file.aiquickdraw.com/mathbot/file/1727058558152ewna8hwb.jpg

The image you uploaded contains the following mathematical statement and question:

**Irrational Rationalization:**  
"We know that \( \sqrt{2} \) is irrational. Therefore, \( \frac{3\sqrt{2}}{5\sqrt{2}} \) must also be irrational. Is this conclusion correct? Why or why not?"

---

### Answer:
No, the conclusion is incorrect. Here's why:

When simplifying the expression \( \frac{3\sqrt{2}}{5\sqrt{2}} \), you can cancel out the \( \sqrt{2} \) in both the numerator and denominator:

\[
\frac{3\sqrt{2}}{5\sqrt{2}} = \frac{3}{5}
\]

Now, \( \frac{3}{5} \) is a rational number because it is a ratio of two integers. Therefore, even though \( \sqrt{2} \) is irrational, the result of this particular expression simplifies to a rational number.

---

Would you like further clarification on any part of this solution?

Here are 5 related questions that might help extend your understanding:

1. What does it mean for a number to be irrational?
2. How can irrational numbers be combined with rational numbers?
3. What are examples of operations that result in rational numbers despite involving irrational numbers?
4. How do you prove that \( \sqrt{2} \) is irrational?
5. Can two irrational numbers be divided to result in a rational number? Under what conditions?

**Tip:** Whenever simplifying fractions with irrational numbers, check if the irrational parts cancel out. This often simplifies the problem!

We know that √2 is irrational. Therefore, 3√2 / 5√2 must also be irrational. Is this conclusion correct? Why or why not?

Learn how to determine whether the expression 3√2 / 5√2 is irrational. This problem covers key concepts such as rational and irrational numbers, and fraction simplification. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation