Obtain the volume of rectangular boxes with the following length, breadth and height respectively. (i) `5a, 3a^2, 7a^4` (ii) `2p, 4q, 8r` (iii) `xy, 2x^2y, 2xy^2` (iv) `a, 2b, 3c`

Explanation

To find the volume of rectangular boxes given their dimensions, we use the formula for the volume of a rectangular box, which is:

\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]

Now, let's calculate the volume for each part step by step.

### (i) Length = \(5a\), Breadth = \(3a^2\), Height = \(7a^4\)

1. **Write the formula for volume**:
\[ V = L \times B \times H \]
\[ V = 5a \times 3a^2 \times 7a^4 \]

2. **Multiply the coefficients**:
\[ 5 \times 3 \times 7 = 105 \]

3. **Multiply the variables**:
\[ a \times a^2 \times a^4 = a^{1+2+4} = a^7 \]

4. **Combine the results**:
\[ V = 105a^7 \]

### (ii) Length = \(2p\), Breadth = \(4q\), Height = \(8r\)

1. **Write the formula for volume**:
\[ V = L \times B \times H \]
\[ V = 2p \times 4q \times 8r \]

2. **Multiply the coefficients**:
\[ 2 \times 4 \times 8 = 64 \]

3. **Combine the variables**:
\[ V = 64pqr \]

### (iii) Length = \(xy\), Breadth = \(2x^2y\), Height = \(2xy^2\)

1. **Write the formula for volume**:
\[ V = L \times B \times H \]
\[ V = xy \times 2x^2y \times 2xy^2 \]

2. **Multiply the coefficients**:
\[ 1 \times 2 \times 2 = 4 \]

3. **Multiply the variables**:
\[ x^1 \times x^2 \times x^1 = x^{1+2+1} = x^4 \]
\[ y^1 \times y^1 \times y^2 = y^{1+1+2} = y^4 \]

4. **Combine the results**:
\[ V = 4x^4y^4 \]

### (iv) Length = \(a\), Breadth = \(2b\), Height = \(3c\)

1. **Write the formula for volume**:
\[ V = L \times B \times H \]
\[ V = a \times 2b \times 3c \]

2. **Multiply the coefficients**:
\[ 1 \times 2 \times 3 = 6 \]

3. **Combine the variables**:
\[ V = 6abc \]

### Final Answers:
1. \( V = 105a^7 \)
2. \( V = 64pqr \)
3. \( V = 4x^4y^4 \)
4. \( V = 6abc \)