Class 8 Maths Cubes and Cube Roots Worksheet - Free Printable

Let's solve each of these problems step by step.

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1. Find the cube of 15

Cube of a number = $ \text{number}^3 $

$$
15^3 = 15 \times 15 \times 15 = 225 \times 15 = 3375
$$

✔ Answer: 3375

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2. Is 3087 a perfect cube?

We check if 3087 is a perfect cube by prime factorization.

Factorize 3087:

- 3087 ÷ 3 = 1029
- 1029 ÷ 3 = 343
- 343 = $ 7^3 $

So:
$$
3087 = 3^2 \times 7^3
$$

For a number to be a perfect cube, all exponents in prime factorization must be multiples of 3.

Here, $ 3^2 $ → exponent 2 (not divisible by 3)

✘ Not a perfect cube

✔ Answer: No

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3. If $ 72x $ is a perfect cube, then $ x = $?

First, factorize 72:

$$
72 = 8 \times 9 = 2^3 \times 3^2
$$

So,
$$
72x = 2^3 \times 3^2 \times x
$$

To make this a perfect cube, all exponents must be multiples of 3.

- $ 2^3 $ → already good
- $ 3^2 $ → needs one more 3 → need $ 3^1 $
- So $ x $ must provide $ 3^1 $, and possibly other factors?

But wait — we want the smallest such $ x $. So only add what’s missing.

So $ x = 3 $

Check:
$$
72 \times 3 = 216 = 6^3 \quad (\text{Yes!})
$$

✔ Answer: 3

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4. Write ones digit of cube root of 2197

We need the ones digit of the cube root of 2197.

First, estimate cube root:

- $ 10^3 = 1000 $
- $ 13^3 = 13 \times 13 \times 13 = 169 \times 13 = 2197 $

Yes! $ 13^3 = 2197 $

So cube root of 2197 is 13 → ones digit is 3

✔ Answer: 3

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5. 17 is a cube root of ________

If 17 is the cube root of a number, then that number is:

$$
17^3 = 17 \times 17 \times 17 = 289 \times 17
$$

Calculate:
$$
289 \times 17 = (300 - 11) \times 17 = 300 \times 17 - 11 \times 17 = 5100 - 187 = 4913
$$

✔ Answer: 4913

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6. The cube root of $ 2^3 \times 3^3 \times 5^3 $ is ________

$$
\sqrt[3]{2^3 \times 3^3 \times 5^3} = 2^{3/3} \times 3^{3/3} \times 5^{3/3} = 2 \times 3 \times 5 = 30
$$

✔ Answer: 30

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7. Find the cube root of:

#### (i) 729

$$
\sqrt[3]{729} = 9 \quad \text{(since } 9^3 = 729)
$$

✔ Answer: 9

#### (ii) $ 64 \times 27 $

$$
64 = 4^3, \quad 27 = 3^3 \Rightarrow \sqrt[3]{64 \times 27} = \sqrt[3]{4^3 \times 3^3} = 4 \times 3 = 12
$$

✔ Answer: 12

#### (iii) 2744

Try estimating:
- $ 10^3 = 1000 $
- $ 14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744 $

Yes! So $ \sqrt[3]{2744} = 14 $

✔ Answer: 14

#### (iv) 64000

Note: $ 64000 = 64 \times 1000 = 4^3 \times 10^3 $

So cube root = $ 4 \times 10 = 40 $

✔ Answer: 40

#### (v) 1.331

Note: $ 1.1^3 = 1.1 \times 1.1 \times 1.1 = 1.21 \times 1.1 = 1.331 $

So $ \sqrt[3]{1.331} = 1.1 $

✔ Answer: 1.1

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8. Find the cube root of $ 140 \times 2450 $

First compute the product:

But better to simplify before multiplying.

Factor both numbers:

- $ 140 = 14 \times 10 = 2 \times 7 \times 2 \times 5 = 2^2 \times 5 \times 7 $
- $ 2450 = 245 \times 10 = (49 \times 5) \times (2 \times 5) = 7^2 \times 5 \times 2 \times 5 = 2 \times 5^2 \times 7^2 $

Now multiply:

$$
140 \times 2450 = (2^2 \times 5 \times 7) \times (2 \times 5^2 \times 7^2) = 2^{3} \times 5^{3} \times 7^{3}
$$

Now take cube root:

$$
\sqrt[3]{2^3 \times 5^3 \times 7^3} = 2 \times 5 \times 7 = 70
$$

✔ Answer: 70

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9. Find the smallest number by which 256 must be multiplied so that the product is a perfect cube

Factor 256:

$$
256 = 2^8
$$

We want $ 2^8 \times x $ to be a perfect cube.

Exponent 8 → need exponent to be multiple of 3.

Next multiple of 3 after 8 is 9 → need one more 2.

So $ x = 2 $

Check: $ 256 \times 2 = 512 = 8^3 = 2^9 $ → yes!

✔ Answer: 2

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10. If $ x^3 = 0.008 $, then $ x = $?

$$
x = \sqrt[3]{0.008} = \sqrt[3]{8 \times 10^{-3}} = \sqrt[3]{8} \times \sqrt[3]{10^{-3}} = 2 \times 10^{-1} = 0.2
$$

✔ Answer: 0.2

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11. Find the side of the cubical box whose volume is 9261 m³

Volume of cube = $ s^3 $

So:
$$
s = \sqrt[3]{9261}
$$

Try:
- $ 20^3 = 8000 $
- $ 21^3 = 21 \times 21 \times 21 = 441 \times 21 = 9261 $

Yes! So $ s = 21 $

✔ Answer: 21 m

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12. If $ a^3 - 1 = 1330 $, then $ a = $?

$$
a^3 = 1330 + 1 = 1331
$$

Now $ \sqrt[3]{1331} = 11 $, since $ 11^3 = 1331 $

✔ Answer: 11

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13. Find the cube root of $ \frac{1331}{1728} $

We know:
- $ 1331 = 11^3 $
- $ 1728 = 12^3 $

So:
$$
\sqrt[3]{\frac{1331}{1728}} = \frac{\sqrt[3]{1331}}{\sqrt[3]{1728}} = \frac{11}{12}
$$

✔ Answer: $ \frac{11}{12} $

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14. If $ \sqrt[3]{\frac{x}{y}} = \frac{2}{3} $, then $ \frac{x}{y} = $?

Cube both sides:

$$
\left( \sqrt[3]{\frac{x}{y}} \right)^3 = \left( \frac{2}{3} \right)^3 \Rightarrow \frac{x}{y} = \frac{8}{27}
$$

✔ Answer: $ \frac{8}{27} $

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15. Three numbers are in the ratio 1 : 2 : 3 and the sum of their cubes is 288. Find the numbers.

Let the numbers be $ x, 2x, 3x $

Sum of cubes:
$$
x^3 + (2x)^3 + (3x)^3 = x^3 + 8x^3 + 27x^3 = 36x^3
$$

Given:
$$
36x^3 = 288 \Rightarrow x^3 = \frac{288}{36} = 8 \Rightarrow x = \sqrt[3]{8} = 2
$$

So the numbers are:
- $ x = 2 $
- $ 2x = 4 $
- $ 3x = 6 $

Check:
$ 2^3 + 4^3 + 6^3 = 8 + 64 + 216 = 288 $ ✔

✔ Answer: 2, 4, 6

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## ✔ Final Answers Summary:

1. 3375
2. No
3. 3
4. 3
5. 4913
6. 30
7. (i) 9, (ii) 12, (iii) 14, (iv) 40, (v) 1.1
8. 70
9. 2
10. 0.2
11. 21 m
12. 11
13. $ \frac{11}{12} $
14. $ \frac{8}{27} $
15. 2, 4, 6

Let me know if you'd like explanations in simpler terms or visual steps!