Find the fourth proportional to :
(i) `1.5`, `4.5` and `3.5`
(ii) `3a, `6a^(2)` and `2ab^(2)`

Answer

To find the fourth proportional to the given numbers, we can use the property of proportions. The fourth proportional \( x \) to three numbers \( a, b, c \) can be found using the formula:

\[ \frac{a}{b} = \frac{c}{x} \]

From this, we can derive that:

\[ x = \frac{b \cdot c}{a} \]

Let's solve the two parts of the question step by step.

### Part (i): Find the fourth proportional to \( 1.5, 4.5, \) and \( 3.5 \)

1. **Assign the values**: Let \( a = 1.5 \), \( b = 4.5 \), and \( c = 3.5 \).
2. **Set up the proportion**: According to the property of proportions, we have:
\[ \frac{1.5}{4.5} = \frac{3.5}{x} \]
3. **Cross-multiply**: This gives us:
\[ 1.5 \cdot x = 4.5 \cdot 3.5 \]
4. **Calculate \( 4.5 \cdot 3.5 \)**:
\[ 4.5 \cdot 3.5 = 15.75 \]
5. **Substitute back**: Now we have:
\[ 1.5 \cdot x = 15.75 \]
6. **Solve for \( x \)**:
\[ x = \frac{15.75}{1.5} = 10.5 \]

Thus, the fourth proportional to \( 1.5, 4.5, \) and \( 3.5 \) is **10.5**.

### Part (ii): Find the fourth proportional to \( 3a, 6a^2, \) and \( 2ab^2 \)

1. **Assign the values**: Let \( a = 3a \), \( b = 6a^2 \), and \( c = 2ab^2 \).
2. **Set up the proportion**: According to the property of proportions, we have:
\[ \frac{3a}{6a^2} = \frac{2ab^2}{x} \]
3. **Cross-multiply**: This gives us:
\[ 3a \cdot x = 6a^2 \cdot 2ab^2 \]
4. **Calculate \( 6a^2 \cdot 2ab^2 \)**:
\[ 6a^2 \cdot 2ab^2 = 12a^3b^2 \]
5. **Substitute back**: Now we have:
\[ 3a \cdot x = 12a^3b^2 \]
6. **Solve for \( x \)**:
\[ x = \frac{12a^3b^2}{3a} = 4a^2b^2 \]

Thus, the fourth proportional to \( 3a, 6a^2, \) and \( 2ab^2 \) is **\( 4a^2b^2 \)**.

### Summary of Answers:
1. The fourth proportional to \( 1.5, 4.5, 3.5 \) is **10.5**.
2. The fourth proportional to \( 3a, 6a^2, 2ab^2 \) is **\( 4a^2b^2 \)**.