which of the following is an irrational number. a)2 root 49  b)root 225  c)2root3-root 12   d) 7root7

To determine which of these options is an irrational number, let's analyze each one individually:

1. **Option (a): \( 2 \sqrt{49} \)**
   - Since \( \sqrt{49} = 7 \), this becomes \( 2 \times 7 = 14 \), which is a rational number.

2. **Option (b): \( \sqrt{225} \)**
   - \( \sqrt{225} = 15 \), which is also a rational number.

3. **Option (c): \( 2\sqrt{3} - \sqrt{12} \)**
   - \( 2\sqrt{3} \) remains irrational because \( \sqrt{3} \) is irrational.
   - \( \sqrt{12} \) simplifies to \( 2\sqrt{3} \).
   - Thus, \( 2\sqrt{3} - 2\sqrt{3} = 0 \), a rational number.

4. **Option (d): \( 7\sqrt{7} \)**
   - Here, \( \sqrt{7} \) is an irrational number, and multiplying it by 7 still results in an irrational number.

**Conclusion:** The irrational number among the options is **(d) \( 7\sqrt{7} \)**.

Would you like further details, or have any questions?

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Here are 5 additional questions for further practice:
1. Is the product of two irrational numbers always irrational?
2. Can the sum of a rational and an irrational number ever be rational?
3. Simplify \( \sqrt{48} + \sqrt{75} \).
4. Explain why \( \sqrt{2} + \sqrt{3} \) is irrational.
5. Determine if \( \sqrt{100} + 5 \) is rational or irrational.

**Tip:** When dealing with square roots, remember that the square root of a perfect square is always rational.

which of the following is an irrational number. a) 2 root 49  b) root 225  c) 2root3-root 12   d) 7root7

Explore how to determine the irrational number from options involving square roots, such as 2√49, √225, 2√3 - √12, and 7√7. This problem covers the properties of irrational numbers and the simplification of radicals, suitable for students in Grades 9-10.

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