To Be Rational or Not To Be Rational - That is the Question ... - Free Printable

Let's solve each problem step by step and determine whether the square root of each number is rational or irrational.

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🔍 Definitions:

- A rational number can be expressed as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \ne 0 $. It has a terminating or repeating decimal.
- An irrational number cannot be written as a simple fraction. Its decimal form never ends or repeats.

For square roots:
- If a number is a perfect square, its square root is rational.
- If it’s not a perfect square, the square root is usually irrational (unless it simplifies to a rational number).

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Now let’s go through each one:

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1. $ \sqrt{16} $

- $ 4^2 = 16 $
- So, $ \sqrt{16} = 4 $
- 4 is an integer → Rational

✔ Answer: Rational

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2. $ \sqrt{24} $

- Is 24 a perfect square? No.
- $ \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} $
- $ \sqrt{6} $ is irrational → so $ 2\sqrt{6} $ is irrational

✔ Answer: Irrational

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3. $ \sqrt{16} $ (again)

- Same as #1 → $ \sqrt{16} = 4 $

✔ Answer: Rational

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4. $ \sqrt{87} $

- 87 is not a perfect square.
- Check between $ 9^2 = 81 $ and $ 10^2 = 100 $
- $ \sqrt{87} \approx 9.327 $ → non-repeating, non-terminating
- Not a perfect square → irrational

✔ Answer: Irrational

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5. $ \sqrt{64} $

- $ 8^2 = 64 $
- So $ \sqrt{64} = 8 $

✔ Answer: Rational

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6. $ \sqrt{32} $

- $ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} $
- $ \sqrt{2} $ is irrational → so this is irrational

✔ Answer: Irrational

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7. $ \sqrt{181} $

- Check: $ 13^2 = 169 $, $ 14^2 = 196 $
- 181 is between them, not a perfect square
- So $ \sqrt{181} $ is irrational

✔ Answer: Irrational

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8. $ \sqrt{121} $

- $ 11^2 = 121 $
- So $ \sqrt{121} = 11 $

✔ Answer: Rational

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9. $ \sqrt{6} $

- 6 is not a perfect square
- $ \sqrt{6} \approx 2.449... $ → non-repeating, non-terminating
- Irrational

✔ Answer: Irrational

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10. $ \sqrt{144} $

- $ 12^2 = 144 $
- So $ \sqrt{144} = 12 $

✔ Answer: Rational

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✔ Final Answers:

| Number | Square Root | Rational or Irrational |
|--------|-------------|-------------------------|
| 1. $ \sqrt{16} $ | 4 | Rational |
| 2. $ \sqrt{24} $ | ≈4.899 | Irrational |
| 3. $ \sqrt{16} $ | 4 | Rational |
| 4. $ \sqrt{87} $ | ≈9.327 | Irrational |
| 5. $ \sqrt{64} $ | 8 | Rational |
| 6. $ \sqrt{32} $ | ≈5.657 | Irrational |
| 7. $ \sqrt{181} $ | ≈13.454 | Irrational |
| 8. $ \sqrt{121} $ | 11 | Rational |
| 9. $ \sqrt{6} $ | ≈2.449 | Irrational |
| 10. $ \sqrt{144} $ | 12 | Rational |

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📝 Summary:

- Rational: 1, 3, 5, 8, 10
- Irrational: 2, 4, 6, 7, 9

Let me know if you'd like this in printable format!