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:

**Volume = Length × Breadth × Height**

Let's solve each part step by step.

### Part (i): Length = 5a, Breadth = 3a², Height = 7a⁴

1. **Write the formula for volume**:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]
\[ V = 5a \times 3a^2 \times 7a^4 \]

2. **Multiply the coefficients (numerical values)**:
\[ V = (5 \times 3 \times 7) \times (a \times a^2 \times a^4) \]
\[ V = 105 \times (a^{1+2+4}) \]

3. **Combine the exponents**:
\[ V = 105a^7 \]

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

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

2. **Multiply the coefficients**:
\[ V = (2 \times 4 \times 8) \times (p \times q \times r) \]
\[ V = 64 \times pqr \]

### Part (iii): Length = xy, Breadth = 2x²y, Height = 2xy²

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

2. **Multiply the coefficients**:
\[ V = (1 \times 2 \times 2) \times (x \times x^2 \times x) \times (y \times y \times y^2) \]
\[ V = 4 \times (x^{1+2+1}) \times (y^{1+1+2}) \]

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

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

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

2. **Multiply the coefficients**:
\[ V = (1 \times 2 \times 3) \times (a \times b \times c) \]
\[ V = 6abc \]

### Summary of Volumes:

1. Part (i): \( V = 105a^7 \)
2. Part (ii): \( V = 64pqr \)
3. Part (iii): \( V = 4x^4y^4 \)
4. Part (iv): \( V = 6abc \)

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