CBSE Class 9 Mental Maths Number System Worksheet - Free Printable

Let's solve each problem step-by-step from the given worksheet on Number System.

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1. Simplify $ \sqrt[4]{32} $

We know:
$$
32 = 2^5
$$
So,
$$
\sqrt[4]{32} = (2^5)^{1/4} = 2^{5/4}
$$
This can be written as:
$$
2^{5/4} = 2^{1 + 1/4} = 2 \cdot 2^{1/4} = 2\sqrt[4]{2}
$$

✔ Answer: $ 2\sqrt[4]{2} $

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2. Which is greater: $ \sqrt[4]{3} $ and $ \sqrt[4]{5} $?

Since the root index (4) is the same, compare the radicands:

- $ 3 < 5 $
- So $ \sqrt[4]{3} < \sqrt[4]{5} $

✔ Answer: $ \sqrt[4]{5} $ is greater.

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3. Arrange in descending order: $ \sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{4} $?

Cube roots increase as the number increases.

- $ 2 < 3 < 4 $
- So $ \sqrt[3]{2} < \sqrt[3]{3} < \sqrt[3]{4} $

Descending order:
$$
\sqrt[3]{4}, \sqrt[3]{3}, \sqrt[3]{2}
$$

✔ Answer: $ \sqrt[3]{4}, \sqrt[3]{3}, \sqrt[3]{2} $

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4. Simplify $ \sqrt{8} + \sqrt{32} - \sqrt{2} $

Break down each square root:
- $ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} $
- $ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} $
- $ \sqrt{2} = \sqrt{2} $

Now:
$$
2\sqrt{2} + 4\sqrt{2} - \sqrt{2} = (2 + 4 - 1)\sqrt{2} = 5\sqrt{2}
$$

✔ Answer: $ 5\sqrt{2} $

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5. Multiply $ 3\sqrt{7} $ by $ \sqrt{2} $

$$
3\sqrt{7} \cdot \sqrt{2} = 3 \cdot \sqrt{7 \cdot 2} = 3\sqrt{14}
$$

✔ Answer: $ 3\sqrt{14} $

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6. Divide $ \sqrt{24} $ by $ \sqrt{200} $

Use rule: $ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $

$$
\frac{\sqrt{24}}{\sqrt{200}} = \sqrt{\frac{24}{200}} = \sqrt{\frac{3}{25}} = \frac{\sqrt{3}}{5}
$$

✔ Answer: $ \frac{\sqrt{3}}{5} $

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7. What is the value of $ \left(8^{3/5}\right)^5 $?

Use exponent rule: $ (a^m)^n = a^{m \cdot n} $

$$
\left(8^{3/5}\right)^5 = 8^{(3/5) \cdot 5} = 8^3 = 512
$$

✔ Answer: $ 512 $

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8. What is the value of $ 4^{1/5} \times 8^{1/5} $?

Note: $ 4 = 2^2 $, $ 8 = 2^3 $

So:
$$
4^{1/5} = (2^2)^{1/5} = 2^{2/5}, \quad 8^{1/5} = (2^3)^{1/5} = 2^{3/5}
$$

Now multiply:
$$
2^{2/5} \cdot 2^{3/5} = 2^{(2+3)/5} = 2^{1} = 2
$$

✔ Answer: $ 2 $

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9. What is the value of $ 8^{1/5} \div 8^{1/3} $?

Use rule: $ a^m \div a^n = a^{m-n} $

$$
8^{1/5 - 1/3} = 8^{(3 - 5)/15} = 8^{-2/15}
$$

Now $ 8 = 2^3 $, so:
$$
(2^3)^{-2/15} = 2^{-6/15} = 2^{-2/5} = \frac{1}{2^{2/5}}
$$

Alternatively, leave as $ 8^{-2/15} $

But simplest form is:
$$
\boxed{8^{-2/15}} \quad \text{or} \quad \boxed{\frac{1}{8^{2/15}}}
$$

✔ Answer: $ 8^{-2/15} $

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10. Simplify $ 6^3 \times \left(\frac{1}{2}\right)^3 $

$$
6^3 = 216, \quad \left(\frac{1}{2}\right)^3 = \frac{1}{8}
$$

So:
$$
216 \times \frac{1}{8} = \frac{216}{8} = 27
$$

✔ Answer: $ 27 $

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11. Simplify $ \frac{3}{4} \sqrt{32} $

First simplify $ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} $

Then:
$$
\frac{3}{4} \cdot 4\sqrt{2} = 3\sqrt{2}
$$

✔ Answer: $ 3\sqrt{2} $

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12. Write into simplest form $ \sqrt[3]{3125} $

Factor 3125:
- $ 3125 \div 5 = 625 $
- $ 625 \div 5 = 125 $
- $ 125 \div 5 = 25 $
- $ 25 \div 5 = 5 $
- $ 5 \div 5 = 1 $

So $ 3125 = 5^5 $

Now:
$$
\sqrt[3]{5^5} = 5^{5/3} = 5^{1 + 2/3} = 5 \cdot 5^{2/3} = 5\sqrt[3]{25}
$$

✔ Answer: $ 5\sqrt[3]{25} $

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13. Simplify $ 4\sqrt{3} + \sqrt{27} $

Simplify $ \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} $

So:
$$
4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}
$$

✔ Answer: $ 7\sqrt{3} $

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14. Find two rational numbers between $ \frac{1}{2} $ and $ \frac{1}{4} $

Note: $ \frac{1}{2} = 0.5 $, $ \frac{1}{4} = 0.25 $

So we need numbers between 0.25 and 0.5.

Let’s find common denominator: $ \frac{1}{2} = \frac{2}{4} $, $ \frac{1}{4} = \frac{1}{4} $

Between $ \frac{1}{4} $ and $ \frac{2}{4} $, try:
- $ \frac{3}{8} = 0.375 $
- $ \frac{1}{3} \approx 0.333 $

Both are between 0.25 and 0.5.

✔ Answer: $ \frac{3}{8}, \frac{1}{3} $ (any two rational numbers in between)

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15. Find two rational numbers between $ -1 $ and $ \frac{3}{2} $

$ -1 = -1.0 $, $ \frac{3}{2} = 1.5 $

Any rational numbers in between:
- $ 0 $
- $ \frac{1}{2} = 0.5 $
- $ \frac{1}{4} = 0.25 $
- $ -\frac{1}{2} = -0.5 $

Choose any two:
✔ Answer: $ 0, \frac{1}{2} $ (or any two values between -1 and 1.5)

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16. Find three rational numbers between $ -5 $ and $ \frac{3}{4} $

$ -5 = -5.0 $, $ \frac{3}{4} = 0.75 $

Pick three numbers between them:
- $ -2 $
- $ 0 $
- $ \frac{1}{2} = 0.5 $

✔ Answer: $ -2, 0, \frac{1}{2} $ (or any three rational numbers in between)

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## ✔ Final Answers Summary:

| Q | Answer |
|---|--------|
| 1 | $ 2\sqrt[4]{2} $ |
| 2 | $ \sqrt[4]{5} $ is greater |
| 3 | $ \sqrt[3]{4}, \sqrt[3]{3}, \sqrt[3]{2} $ |
| 4 | $ 5\sqrt{2} $ |
| 5 | $ 3\sqrt{14} $ |
| 6 | $ \frac{\sqrt{3}}{5} $ |
| 7 | $ 512 $ |
| 8 | $ 2 $ |
| 9 | $ 8^{-2/15} $ or $ \frac{1}{8^{2/15}} $ |
|10 | $ 27 $ |
|11 | $ 3\sqrt{2} $ |
|12 | $ 5\sqrt[3]{25} $ |
|13 | $ 7\sqrt{3} $ |
|14 | $ \frac{3}{8}, \frac{1}{3} $ (example) |
|15 | $ 0, \frac{1}{2} $ (example) |
|16 | $ -2, 0, \frac{1}{2} $ (example) |

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