Question

Find the following product: \[\left( - \frac{7}{4}a b^2 c - \frac{6}{25} a^2 c^2 \right)( - 50 a^2 b^2 c^2 )\]

Answer 1

To find the product, we will use distributive law as follows:

\[\left( - \frac{7}{4}a b^2 c - \frac{6}{25} a^2 c^2 \right)\left( - 50 a^2 b^2 c^2 \right)\]

\[ = \left\{ \left( - \frac{7}{4}a b^2 c \right)\left( - 50 a^2 b^2 c^2 \right) \right\} - \left\{ \left( \frac{6}{25} a^2 c^2 \right)\left( - 50 a^2 b^2 c^2 \right) \right\}\]

\[ = \left\{ \left\{ - \frac{7}{4} \times \left( - 50 \right) \right\}\left( a \times a^2 \right) \times \left( b^2 \times b^2 \right) \times \left( c \times c^2 \right) \right\} - \left\{ \left( \frac{6}{25} \right)\left( - 50 \right)\left( a^2 \times a^2 \right) \times \left( b^2 \right) \times \left( c^2 \times c^2 \right) \right\}\]

\[ = \left\{ - \frac{7}{4} \times \left( - 50 \right) \right\}\left( a^{1 + 2} b^{2 + 2} c^{1 + 2} \right) - \left\{ \left( \frac{6}{25} \right)\left( - 50 \right)\left( a^{2 + 2} b^2 c^{2 + 2} \right) \right\}\]

\[ = \frac{175}{2} a^3 b^4 c^3 - \left( - 12 a^4 b^2 c^4 \right)\]

\[ = \frac{175}{2} a^3 b^4 c^3 + 12 a^4 b^2 c^4\]

Thus, the answer is \[\frac{175}{2} a^3 b^4 c^3 + 12 a^4 b^2 c^4\].