Let's break down the questions from the image:

1. Evaluate:

• i) $(21)^3$
• ii) $(0.05)^3$
• iii) $\left(\frac{10}{11}\right)^3$

Solution: i) $(21)^3 = 21 \times 21 \times 21 = 9261$ ii) $(0.05)^3 = (5 \times 10^{-2})^3 = 5^3 \times 10^{-6} = 125 \times 10^{-6} = 0.000125$ iii) $\left(\frac{10}{11}\right)^3 = \frac{10^3}{11^3} = \frac{1000}{1331}$

2. Show that 3375 is a perfect cube. Find the number whose cube is 3375.

Solution: To show that 3375 is a perfect cube, we need to find the cube root of 3375.

$\sqrt[3]{3375} = 15$ (since $15^3 = 3375$).

So, 3375 is a perfect cube and the number whose cube is 3375 is 15.

3. Find the smallest number by which 2560 must be multiplied so that the product is a perfect cube.

Solution: We can factorize 2560: $2560 = 2^8 \times 5$

To make it a perfect cube, we need to ensure that each prime factor has an exponent that's a multiple of 3:

• For $2^8$, we need one more $2^1$ to make it $2^9$ (which is a perfect cube).
• For $5^1$, we need two more $5^2$ to make it $5^3$ (which is a perfect cube).

So, multiply by $2^1 \times 5^2 = 2 \times 25 = 50$.

Thus, the smallest number is 50.

4. Find the smallest number by which 8788 must be divided so that the quotient is a perfect cube.

Solution: First, factorize 8788: $8788 = 2^2 \times 7^2 \times 13$

For it to become a perfect cube, we need:

• $2^2$ needs one more $2^1$ to make it $2^3$,
• $7^2$ needs one more $7^1$ to make it $7^3$,
• $13^1$ needs two more $13^2$ to make it $13^3$.

Thus, the smallest number we need to divide by is $2^1 \times 7^1 \times 13^2 = 2 \times 7 \times 169 = 236$.

Thus, divide by 236.

5. Evaluate:

$\sqrt[3]{288} \times \frac{72}{27}$

Solution:

• $\sqrt[3]{288} \approx 6.643$,
• $\frac{72}{27} = \frac{8}{3}$.

Multiplying these together:

$6.643 \times \frac{8}{3} \approx 17.709.$

Would you like more details on any of these problems? Here are a few related questions that might help:

1. How do you find the cube of a number algebraically?
2. What are perfect cubes, and how can you recognize them?
3. How can you determine the cube root of a large number without a calculator?
4. How does prime factorization help in identifying perfect cubes?
5. Can a non-perfect cube be converted into a perfect cube? If so, how?

Tip: When dealing with cubes and cube roots, always consider prime factorization to break down the problem into simpler parts!