Question

Find the following product: \[- \frac{8}{27}xyz\left( \frac{3}{2}xy z^2 - \frac{9}{4}x y^2 z^3 \right)\]

Answer 1

To find the product, we will use the distributive law in the following way:

\[- \frac{8}{27}xyz\left( \frac{3}{2}xy z^2 - \frac{9}{4}x y^2 z^3 \right)\]

\[ = \left\{ \left( - \frac{8}{27}xyz \right)\left( \frac{3}{2}xy z^2 \right) \right\} - \left\{ \left( - \frac{8}{27}xyz \right)\left( \frac{9}{4}x y^2 z^3 \right) \right\}\]

\[ = \left\{ \left( - \frac{8}{27} \times \frac{3}{2} \right)\left( x \times x \right) \times \left( y \times y \right) \times \left( z \times z^2 \right) \right\} - \left\{ \left( - \frac{8}{27} \times \frac{9}{4} \right)\left( x \times x \right) \times \left( y \times y^2 \right) \times \left( z \times z^3 \right) \right\}\]

\[ = \left\{ \left( - \frac{8}{27} \times \frac{3}{2} \right)\left( x^{1 + 1} y^{1 + 1} z^{1 + 2} \right) \right\} - \left\{ \left( - \frac{8}{27} \times \frac{9}{4} \right)\left( x^{1 + 1} y^{1 + 2} z^{1 + 3} \right) \right\}\]

\[ = \left\{ \left( - \frac{8^4}{{27}_9} \times \frac{3}{2} \right)\left( x^{1 + 1} y^{1 + 1} z^{1 + 2} \right) \right\} - \left\{ \left( - \frac{8^2}{{27}_3} \times \frac{9}{4} \right)\left( x^{1 + 1} y^{1 + 2} z^{1 + 3} \right) \right\}\]

\[ = - \frac{4}{9} x^2 y^2 z^3 + \frac{2}{3} x^2 y^3 z^4\]

Thus, the answer is \[- \frac{4}{9} x^2 y^2 z^3 + \frac{2}{3} x^2 y^3 z^4\].