Let’s go through each of the questions and solve them one by one:

1. Find the cube root of 6859.

To find the cube root of 6859:

$\sqrt[3]{6859} = 19$

2. Find the cube root of 13824 through estimation.

Since $20^3 = 8000$ and $30^3 = 27000$, the cube root of 13824 lies between 20 and 30. You can estimate that it is around 24.

$\sqrt[3]{13824} = 24$

3. Find the smallest number by which 3087 must be divided so that the quotient is a perfect cube.

First, factorize 3087: $3087 = 3^3 \times 19$ To make it a perfect cube, divide by 19, leaving $3^3$, which is a perfect cube.

4. Is 392 a perfect cube?

Factorizing 392: $392 = 2^3 \times 7$ Since all factors are not perfect cubes, 392 is not a perfect cube.

5. By what least number should 968 be multiplied to get a perfect cube? Find the number whose cube is the new number.

Factorize 968: $968 = 2^3 \times 11$ To make it a perfect cube, multiply by $11^2$. The number becomes $2^3 \times 11^3 = (2 \times 11)^3 = 22^3$.

So, multiplying 968 by $11^2 = 121$, the new number is $22^3 = 10648$.

6. Is the cube of 4913 an odd number? Why?

Since 4913 is an odd number, its cube will also be odd.

7. Is the cube of 132651 an even number? Why?

Since 132651 is an odd number, its cube will also be odd.

8. Find the cubes of these numbers:

• $4^3 = 64$
• $8^3 = 512$
• $15^3 = 3375$
• $0.2^3 = 0.008$
• $1.1^3 = 1.331$

9. Find the cube roots of these numbers:

(i) $4 - \frac{12}{125}$
First simplify the expression $4 - \frac{12}{125} \approx 3.904$, and then: $\sqrt[3]{3.904} \approx 1.571$

(ii) $\sqrt[3]{-0.729} = -0.9$

(iii) $\sqrt[3]{5832} = 18$

(iv) $\sqrt[3]{64000} = 40$

(v) $\sqrt[3]{10648} = 22$

Would you like further details on any of these solutions?

Here are 5 related questions to explore:

1. What is the cube root of 15625?
2. How do you check if a large number is a perfect cube?
3. What is the smallest cube greater than 50000?
4. How do you estimate cube roots using binomial expansion?
5. What is the cube of 25.5?

Tip: Cube roots can often be estimated by checking the cubes of integers near the given number, especially for large numbers.