Write the following cubes in the expanded form :
(i) `(2a+3b)^(3)`
(ii) `(3p-5q)^(3)`

Answer

To expand the given cubes, we will use the formulas for the cube of a binomial:

1. **For \( (a + b)^3 \)**:
\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]

2. **For \( (a - b)^3 \)**:
\[ (a - b)^3 = a^3 - b^3 - 3a^2b + 3ab^2 \]

Now, let's apply these formulas to the given questions.

### (i) Expanding \( (2a + 3b)^3 \)

1. Identify \( a \) and \( b \):
- \( a = 2a \)
- \( b = 3b \)

2. Apply the formula:
\[ (2a + 3b)^3 = (2a)^3 + (3b)^3 + 3(2a)^2(3b) + 3(2a)(3b)^2 \]

3. Calculate each term:
- \( (2a)^3 = 8a^3 \)
- \( (3b)^3 = 27b^3 \)
- \( 3(2a)^2(3b) = 3 \cdot 4a^2 \cdot 3b = 36a^2b \)
- \( 3(2a)(3b)^2 = 3 \cdot 2a \cdot 9b^2 = 54ab^2 \)

4. Combine all the terms:
\[ (2a + 3b)^3 = 8a^3 + 27b^3 + 36a^2b + 54ab^2 \]

### (ii) Expanding \( (3p - 5q)^3 \)

1. Identify \( a \) and \( b \):
- \( a = 3p \)
- \( b = 5q \)

2. Apply the formula:
\[ (3p - 5q)^3 = (3p)^3 - (5q)^3 - 3(3p)^2(5q) + 3(3p)(5q)^2 \]

3. Calculate each term:
- \( (3p)^3 = 27p^3 \)
- \( (5q)^3 = 125q^3 \)
- \( -3(3p)^2(5q) = -3 \cdot 9p^2 \cdot 5q = -135p^2q \)
- \( 3(3p)(5q)^2 = 3 \cdot 3p \cdot 25q^2 = 225pq^2 \)

4. Combine all the terms:
\[ (3p - 5q)^3 = 27p^3 - 125q^3 - 135p^2q + 225pq^2 \]

### Final Answers:

1. \( (2a + 3b)^3 = 8a^3 + 27b^3 + 36a^2b + 54ab^2 \)
2. \( (3p - 5q)^3 = 27p^3 - 125q^3 - 135p^2q + 225pq^2 \)