Find the product :
`(i) (x+3)(x^(2)-3x+9) " " (ii) (7+5b)(49-35b+25b^(2))`
`(iii) (5a+(1)/(2)) (25a^(2)-(5a)/(2)+(1)/(4)) " " `.

Answer

Let's solve the given problems step by step.

### Part (i): Find the product of \((x + 3)(x^2 - 3x + 9)\)

1. **Distribute \(x\) to each term in the second polynomial**:
\[ x \cdot x^2 = x^3 \]
\[ x \cdot (-3x) = -3x^2 \]
\[ x \cdot 9 = 9x \]

2. **Distribute \(3\) to each term in the second polynomial**:
\[ 3 \cdot x^2 = 3x^2 \]
\[ 3 \cdot (-3x) = -9x \]
\[ 3 \cdot 9 = 27 \]

3. **Combine all the terms**:
\[ x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 \]

4. **Combine like terms**:
\[ x^3 + ( -3x^2 + 3x^2) + (9x - 9x) + 27 = x^3 + 27 \]

**Final answer for part (i)**:
\[ \boxed{x^3 + 27} \]

### Part (ii): Find the product of \((7 + 5b)(49 - 35b + 25b^2)\)

1. **Distribute \(7\) to each term in the second polynomial**:
\[ 7 \cdot 49 = 343 \]
\[ 7 \cdot (-35b) = -245b \]
\[ 7 \cdot 25b^2 = 175b^2 \]

2. **Distribute \(5b\) to each term in the second polynomial**:
\[ 5b \cdot 49 = 245b \]
\[ 5b \cdot (-35b) = -175b^2 \]
\[ 5b \cdot 25b^2 = 125b^3 \]

3. **Combine all the terms**:
\[ 343 - 245b + 175b^2 + 245b - 175b^2 + 125b^3 \]

4. **Combine like terms**:
\[ 343 + ( -245b + 245b) + (175b^2 - 175b^2) + 125b^3 = 125b^3 + 343 \]

**Final answer for part (ii)**:
\[ \boxed{125b^3 + 343} \]

### Part (iii): Find the product of \(\left(5a + \frac{1}{2}\right)\left(25a^2 - \frac{5a}{2} + \frac{1}{4}\right)\)

1. **Distribute \(5a\) to each term in the second polynomial**:
\[ 5a \cdot 25a^2 = 125a^3 \]
\[ 5a \cdot \left(-\frac{5a}{2}\right) = -\frac{25a^2}{2} \]
\[ 5a \cdot \frac{1}{4} = \frac{5a}{4} \]

2. **Distribute \(\frac{1}{2}\) to each term in the second polynomial**:
\[ \frac{1}{2} \cdot 25a^2 = \frac{25a^2}{2} \]
\[ \frac{1}{2} \cdot \left(-\frac{5a}{2}\right) = -\frac{5a}{4} \]
\[ \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \]

3. **Combine all the terms**:
\[ 125a^3 - \frac{25a^2}{2} + \frac{5a}{4} + \frac{25a^2}{2} - \frac{5a}{4} + \frac{1}{8} \]

4. **Combine like terms**:
\[ 125a^3 + ( -\frac{25a^2}{2} + \frac{25a^2}{2}) + ( \frac{5a}{4} - \frac{5a}{4}) + \frac{1}{8} = 125a^3 + \frac{1}{8} \]

**Final answer for part (iii)**:
\[ \boxed{125a^3 + \frac{1}{8}} \]