Expand :
`(i) (2x+3y)^(3) " " (ii) (5y-3x)^(3) " " (iii) (2a+3b)^(3)`.

Answer

To expand the given expressions, we will use the identities 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 solve each part step by step.

### (i) Expand \( (2x + 3y)^3 \)

1. Identify \( a = 2x \) and \( b = 3y \).
2. Apply the identity for \( (a + b)^3 \):
\[ (2x + 3y)^3 = (2x)^3 + (3y)^3 + 3(2x)^2(3y) + 3(2x)(3y)^2 \]

3. Calculate each term:
- \( (2x)^3 = 8x^3 \)
- \( (3y)^3 = 27y^3 \)
- \( 3(2x)^2(3y) = 3 \cdot 4x^2 \cdot 3y = 36x^2y \)
- \( 3(2x)(3y)^2 = 3 \cdot 2x \cdot 9y^2 = 54xy^2 \)

4. Combine all terms:
\[ (2x + 3y)^3 = 8x^3 + 27y^3 + 36x^2y + 54xy^2 \]

### (ii) Expand \( (5y - 3x)^3 \)

1. Identify \( a = 5y \) and \( b = 3x \).
2. Apply the identity for \( (a - b)^3 \):
\[ (5y - 3x)^3 = (5y)^3 - (3x)^3 - 3(5y)^2(3x) + 3(5y)(3x)^2 \]

3. Calculate each term:
- \( (5y)^3 = 125y^3 \)
- \( (3x)^3 = 27x^3 \)
- \( -3(5y)^2(3x) = -3 \cdot 25y^2 \cdot 3x = -225y^2x \)
- \( 3(5y)(3x)^2 = 3 \cdot 5y \cdot 9x^2 = 135yx^2 \)

4. Combine all terms:
\[ (5y - 3x)^3 = 125y^3 - 27x^3 - 225y^2x + 135yx^2 \]

### (iii) Expand \( (2a + 3b)^3 \)

1. Identify \( a = 2a \) and \( b = 3b \).
2. Apply the identity for \( (a + b)^3 \):
\[ (2a + 3b)^3 = (2a)^3 + (3b)^3 + 3(2a)^2(3b) + 3(2a)(3b)^2 \]

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 terms:
\[ (2a + 3b)^3 = 8a^3 + 27b^3 + 36a^2b + 54ab^2 \]

### Final Answers:
1. \( (2x + 3y)^3 = 8x^3 + 27y^3 + 36x^2y + 54xy^2 \)
2. \( (5y - 3x)^3 = 125y^3 - 27x^3 - 225y^2x + 135yx^2 \)
3. \( (2a + 3b)^3 = 8a^3 + 27b^3 + 36a^2b + 54ab^2 \)