\( \left. \begin{array} { l } { \left. \begin{array} { l l } { ( 8 x ^ { 3 } ) ( 2 x ^ { 2 } ) = } & { ( - 3 x y ) ( - 2 x ^ { 2 } ) ( - 4 x ^ { 2 } y ) = } \\ { ( - 5 a ) ( 3 b ) = } & { ( 3 x ) ( 3 x ) ( 3 x ) ( 3 x ) = } \\ { ( - 6 x y ) ( - 6 x y ) = } & { ( 2 a ) ( 3 b ) ( 5 c ) = } \\ { ( x ^ { 2 } y z ^ { 3 } ) ( x y ^ { 4 } ) = } & { ( x y ^ { 2 } ) ( x y z ) ( x y z ^ { 3 } ) = } \end{array} \right. } \end{array} \right. \)

Certainly! Let's evaluate each of the given algebraic expressions by multiplying the coefficients and combining like variables. \[ \begin{array}{ll} (8x^3)(2x^2) = 16x^5 & (-3xy)(-2x^2)(-4x^2 y) = -24x^5 y^2 \\ (-5a)(3b) = -15ab & (3x)(3x)(3x)(3x) = 81x^4 \\ (-6xy)(-6xy) = 36x^2 y^2 & (2a)(3b)(5c) = 30abc \\ (x^2 y z^3)(x y^4) = x^3 y^5 z^3 & (x y^2)(x y z)(x y z^3) = x^3 y^4 z^4 \\ \end{array} \] ### Detailed Breakdown: 1. **\((8x^3)(2x^2)\)** - **Coefficients:** \(8 \times 2 = 16\) - **Variables:** \(x^{3+2} = x^5\) - **Result:** \(16x^5\) 2. **\((-3xy)(-2x^2)(-4x^2 y)\)** - **Coefficients:** \((-3) \times (-2) \times (-4) = -24\) - **Variables:** \(x^{1+2+2} = x^5\) and \(y^{1+1} = y^2\) - **Result:** \(-24x^5 y^2\) 3. **\((-5a)(3b)\)** - **Coefficients:** \(-5 \times 3 = -15\) - **Variables:** \(a \times b = ab\) - **Result:** \(-15ab\) 4. **\((3x)(3x)(3x)(3x)\)** - **Coefficients:** \(3 \times 3 \times 3 \times 3 = 81\) - **Variables:** \(x^{1+1+1+1} = x^4\) - **Result:** \(81x^4\) 5. **\((-6xy)(-6xy)\)** - **Coefficients:** \((-6) \times (-6) = 36\) - **Variables:** \(x^{1+1} = x^2\) and \(y^{1+1} = y^2\) - **Result:** \(36x^2 y^2\) 6. **\((2a)(3b)(5c)\)** - **Coefficients:** \(2 \times 3 \times 5 = 30\) - **Variables:** \(a \times b \times c = abc\) - **Result:** \(30abc\) 7. **\((x^2 y z^3)(x y^4)\)** - **Coefficients:** \(1 \times 1 = 1\) (implicit) - **Variables:** - \(x^{2+1} = x^3\) - \(y^{1+4} = y^5\) - \(z^3\) (unchanged) - **Result:** \(x^3 y^5 z^3\) 8. **\((x y^2)(x y z)(x y z^3)\)** - **Coefficients:** \(1 \times 1 \times 1 = 1\) (implicit) - **Variables:** - \(x^{1+1+1} = x^3\) - \(y^{2+1+1} = y^4\) - \(z^{0+1+3} = z^4\) (Note: \(z\) in the first term has an exponent of 0) - **Result:** \(x^3 y^4 z^4\) Feel free to ask if you need further explanations on any of these steps!