Obtain the product of (i) `xy, yz, zx` (ii) `a,-a^2, a^3` (iii) `2, 4y, 8y^2, 16y^3` (iv) `a, 2b, 3c,6abc` (v) `m,-mn, mnp`

Explanation

To solve the question, we need to find the product of the given sets of algebraic expressions step by step.

### (i) Product of `xy, yz, zx`

1. **Write the expression**:
\[ xy \cdot yz \cdot zx \]

2. **Rearrange the terms**:
\[ (x \cdot y) \cdot (y \cdot z) \cdot (z \cdot x) = x \cdot y^2 \cdot z^2 \]

3. **Combine like terms**:
\[ = x^1 \cdot y^{1+1} \cdot z^{1+1} = x^1 \cdot y^2 \cdot z^2 \]

4. **Final result**:
\[ = x^1 y^2 z^2 \]

### (ii) Product of `a, -a^2, a^3`

1. **Write the expression**:
\[ a \cdot (-a^2) \cdot a^3 \]

2. **Rearrange the terms**:
\[ = -1 \cdot a^1 \cdot a^2 \cdot a^3 \]

3. **Combine like terms**:
\[ = -1 \cdot a^{1+2+3} = -a^6 \]

4. **Final result**:
\[ = -a^6 \]

### (iii) Product of `2, 4y, 8y^2, 16y^3`

1. **Write the expression**:
\[ 2 \cdot 4y \cdot 8y^2 \cdot 16y^3 \]

2. **Rearrange the terms**:
\[ = (2 \cdot 4 \cdot 8 \cdot 16) \cdot (y \cdot y^2 \cdot y^3) \]

3. **Calculate the numerical part**:
\[ 2 \cdot 4 = 8, \quad 8 \cdot 8 = 64, \quad 64 \cdot 16 = 1024 \]

4. **Combine like terms**:
\[ = 1024 \cdot y^{1+2+3} = 1024y^6 \]

5. **Final result**:
\[ = 1024y^6 \]

### (iv) Product of `a, 2b, 3c, 6abc`

1. **Write the expression**:
\[ a \cdot 2b \cdot 3c \cdot 6abc \]

2. **Rearrange the terms**:
\[ = (2 \cdot 3 \cdot 6) \cdot (a \cdot a) \cdot (b \cdot b) \cdot (c \cdot c) \]

3. **Calculate the numerical part**:
\[ 2 \cdot 3 = 6, \quad 6 \cdot 6 = 36 \]

4. **Combine like terms**:
\[ = 36 \cdot a^{1+1} \cdot b^{1+1} \cdot c^{1+1} = 36a^2b^2c^2 \]

5. **Final result**:
\[ = 36a^2b^2c^2 \]

### (v) Product of `m, -mn, mnp`

1. **Write the expression**:
\[ m \cdot (-mn) \cdot (mnp) \]

2. **Rearrange the terms**:
\[ = -1 \cdot m \cdot m \cdot n \cdot m \cdot n \cdot p \]

3. **Combine like terms**:
\[ = -1 \cdot m^{1+1+1} \cdot n^{1+1} \cdot p^1 = -m^3n^2p \]

4. **Final result**:
\[ = -m^3n^2p \]

### Summary of Results:
1. \( x^1 y^2 z^2 \)
2. \( -a^6 \)
3. \( 1024y^6 \)
4. \( 36a^2b^2c^2 \)
5. \( -m^3n^2p \)