question_answer 19)

                            Find the product, using suitable properties: (a) \[~\text{26}\times (-\text{ 48})+(-\text{ 48})\times (-\text{ 36})\]                     (b) \[\text{8}\times \text{53}\times (-\text{125})\] (c) \[15\times (-25)\times (-4)\times (-10)\]                        (d) \[(-41)\times 102\] (e) \[\text{625}\times (-\text{35})+(-\text{625})\times \text{65}\]                          (f) \[\text{7}\times (\text{5}0-\text{2})\] (g) \[\text{(}-\text{17)}\times (-\text{ 29})\]                                                      (h) \[(-57)\times (-19)+57\]                

Answer:

                (a)\[26\times (-48)+(-48)\times (-36)\]  \[\text{26}\times (-\text{48})+(-\text{48})\times (-\text{36})\] \[=[-(\text{26}\times \text{48})]+(\text{48}\times \text{36})\] \[\left| \begin{matrix}    a\times (-b)=-(a\times b)  \\    (-a)\times (-b)=a\times b  \\ \end{matrix} \right.\] \[=(-\text{1248})+\text{1728}\] \[\text{= 480}\] Aliter. \[\text{26}\times (-\text{ 48})+(-\text{ 48})\times (-\text{ 36})\] \[=\text{26}\times (-\text{ 48})+(-\text{36})\times (-\text{48})\]                      \[=[\text{26}+(-\text{36})]\times (-\text{48})\]   [Distributive property] \[=(-\text{1}0)\times (-\text{48})\] \[=\text{48}0\]. (b) \[8\times 53\times (-125)\] \[\text{8}\times \text{53}\times (-\text{125})=(\text{8}\times \text{53})\times (-\text{125})\] \[=\text{424}\times (-\text{125})=-\text{ (424}\times \text{125)}\] \[\left| a\times (-b)=-(a\times b) \right.\]               \[=-[\text{424}\times (\text{1}00+\text{25})]=-[\text{424}\times \text{1}00+\text{424}\times \text{25}]\] \[\left| \text{Distributivity of multiplication over addition} \right.\] \[=-[\text{424}00\text{ }+\text{1}0\text{6}00]=-\text{ 53}000\] Aliter. \[\text{8}\times \text{53}\times (-\text{ 125})=\text{8}\times (-\text{ 125})\times \text{53}\]            \[\left| \text{Commutativity of multiplication} \right.\] \[=[\text{8}\times \text{(}-\text{125)}]\times \text{53}\] \[=[-\text{ (8}\times \text{125)}]\times \text{53}\] \[\left| a\times (-b)=-(a\times b) \right.\] \[=(-\text{1}000)\times \text{53}\] \[=-\text{ (1}000\times \text{53)}\] \[\left| (-a)\times b=-(a\times b) \right.\] \[=-\text{53}000.\] (c) \[15\times (-\mathbf{ }25)\times (-\mathbf{ }4)\times (-\mathbf{ }10)\] \[\text{15}\times (-\text{ 25})\times (-\text{ 4})\times (-\text{ 1}0)\] \[=\text{15}\times (-\text{ 25})\times (-\text{ 1}0)\times (-\text{ 4})\]       \[\left| \text{Commutativity of multiplication} \right.\] \[=\text{15}\times (-\text{1}0)\times (-\text{ 25})\times (-\text{ 4})\]       \[\left| \text{Commutativity of multiplication} \right.\] \[=[(\text{15})\times (-\text{l}0)]\times [(-\text{25})\times (-\text{4})]\] \[=[-(15\times 10)]\times [25\times 4]\] \[\left| \begin{matrix}    a\times (-b)=-(a\times b)  \\    (-a)\times (-b)=a\times b  \\ \end{matrix} \right.\]                        \[=(-\text{15}0)\times \text{1}00=-(\text{15}0\text{ x 1}00)=-\text{15}000.\] (d) \[(-41)\times 102\] \[\text{(}-\text{41) }\times \text{1}0\text{2}=-(\text{41}\times \text{1}0\text{2})\] \[\left| (-a)\times b=-(a\times b) \right.\]                            \[=-[\text{41}\times (\text{l}00+\text{2})]\] \[=-[\text{41}\times \text{1}00+\text{41}\times \text{2}]\] \[\left| \text{Distributivity of multiplication over addition} \right.\] \[=-[\text{41}00+\text{82}]=-4\text{182}\]. (e) \[~625\times (-\mathbf{ }35)+(-\mathbf{ }625)\times 65\] \[\text{625}\times (-\text{35})+(-\text{625})\times \text{65}\] \[=\text{625}\times (-\text{ 35})+\text{625}\times (-\text{ 65})\]          \[\left| (-a)\times b=a\times (-b) \right.\]                 \[=\text{625}\times [(-\text{35})+(-\text{65})]\] \[\left| \text{Distributivity of multiplication over addition} \right.\] \[=\text{625}\times (-\text{ 1}00)=-(\text{625}\times \text{1}00)\]       \[\left| a\times (-b)=-(a\times b) \right.\] \[=-\text{ 625}00.\]                 (f) \[7\times [50-2]\] \[\text{7}\times [\text{5}0-\text{2}]=\text{7}\times \text{5}0-\text{7}\times \text{2}\]    \[\left| \text{Distributivity of multiplication over subtraction} \right.\]\[=\text{35}0-\text{14}=\text{336}.\] (g) \[\text{(}-\text{17)}\times (-\text{29})\] \[\text{(}-\text{17)}\times (-\text{ 29})=\text{17}\times \text{29}\] \[\left| (-a)\times (-b)=a\times b \right.\]                                    \[=\text{17}\times (\text{3}0-\text{1})=\text{17}\times \text{3}0-\text{17}\times \text{1}\] \[\left| \text{Distributivity of multiplication over subtraction} \right.\]   \[=\text{51}0-\text{17}=\text{493}.\] (h) \[(-\mathbf{ }57)\times (-19)+57\] \[\text{(}-\text{57)}\times (-\text{19})+\text{57}=\text{57}\times \text{19}+\text{57}\] \[\left| ~(-\text{ }a)\times (-\text{ }b)=a\times b \right.\] \[=\text{57}\times \text{19}+\text{57}\times 1\] \[\left| a\times 1=a \right.\] \[=\text{57}\times (\text{l9}+\text{l})\]        \[\left| \text{Distributivity of multiplication over addition} \right.\] \[=57\times 20=1140\].