question_answer 1)

                Simplify:                 (i) \[{{({{a}^{2}}-{{b}^{2}})}^{2}}\]                 (ii) \[{{(2x+5)}^{2}}-{{(2x-5)}^{2}}\]                 (iii) \[{{(7m-8n)}^{2}}+{{(7m+8n)}^{2}}\]                 (iv) \[{{(4m+5n)}^{2}}+{{(5m+4n)}^{2}}\]                 (v) \[{{(2.5p-1.5q)}^{2}}-{{(1.5p-2.5q)}^{2}}\]                 (vi) \[{{(ab+bc)}^{2}}-2a{{b}^{2}}c\]                 (vii) \[{{({{m}^{2}}-{{n}^{2}}m)}^{2}}\,+2{{m}^{3}}{{n}^{2}}\]S

Answer:

                (i) \[{{({{a}^{2}}-{{b}^{2}})}^{2}}\]            \[{{({{a}^{2}}-{{b}^{2}})}^{2}}\]\[={{({{a}^{2}})}^{2}}-2({{a}^{2}})\,({{b}^{2}})\,+{{({{b}^{2}})}^{2}}\] \[={{a}^{4}}-2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\]                 (ii) \[{{(2x+5)}^{2}}-{{(2x-5)}^{2}}\] \[{{(2x+5)}^{2}}-{{(2x-5)}^{2}}\] \[=\{{{(2x)}^{2}}+2\,(2x)\,(5)\,+{{(5)}^{2}}\}\]\[-\{{{(2x)}^{2}}-2\,(2x)\,(5)\,+{{(5)}^{2}}\}\] \[=(4{{x}^{2}}+20x+25)-(4{{x}^{2}}-20x+25)\] \[=4{{x}^{2}}+20x+25-4{{x}^{2}}+20x-25\] \[=40x\]                 (iii) \[{{(7m-8n)}^{2}}+{{(7m+8n)}^{2}}\]               \[{{(7m-8n)}^{2}}+{{(7m+8n)}^{2}}\] \[=\{{{(7m)}^{2}}\,-2\,(7m)\,(8n)\,+{{(8n)}^{2}}\}\]\[+\{{{(7m)}^{2}}+2\,(7m)\,(8n)\,+{{(8n)}^{2}}\}\]                 \[=(49{{m}^{2}}-112\,mn+64{{n}^{2}})\]\[+(49{{m}^{2}}+112mn+64{{n}^{2}})\] \[=2(49\,{{m}^{2}}+64{{n}^{2}})=98{{m}^{2}}+128{{n}^{2}}\] (iv) \[{{(4m+5n)}^{2}}+{{(5m+4n)}^{2}}\] \[{{(4m+5n)}^{2}}+{{(5m+4n)}^{2}}\] \[=\{{{(4m)}^{2}}+2(4m)\,(5n)\,+{{(5n)}^{2}}\}\]\[+\{{{(5m)}^{2}}+2(5m)\,(4n)+{{(4n)}^{2}}\}\] \[=(16{{m}^{2}}+40mn\,+25{{n}^{2}})\]\[+(25{{m}^{2}}+40mn+16{{n}^{2}})\] \[=(16{{m}^{2}}+25{{m}^{2}})+(40mn+40mn)\]\[+(25{{m}^{2}}+16{{n}^{2}})\] \[=41{{m}^{2}}+80mn+41{{n}^{2}}\]                 (v) \[{{(2.5p-1.5q)}^{2}}-{{(1.5p-2.5q)}^{2}}\]      \[{{(2.5p-1.5q)}^{2}}-{{(1.5p-2.5q)}^{2}}\] \[=\{{{(2.5p)}^{2}}-2\,(2.5p)\,(1.5q)\,+{{(1.5q)}^{2}}\}\] \[-\{{{(1.5p)}^{2}}-2(1.5p)\,(2.5q)\,-{{(2.5q)}^{2}}\}\] \[=(6.2\,5{{p}^{2}}-7.5pq+2.25{{q}^{2}})\]\[-(2.25\,{{p}^{2}}-7.5pq+6.25{{q}^{2}})\] \[=6.25{{p}^{2}}-7.5pq+2.25{{q}^{2}}-2.25{{p}^{2}}\]\[+7.5\,pq-6.25{{q}^{2}}\] \[=(6.25{{p}^{2}}-2.25{{p}^{2}})+(7.5pq-7.5pq)\]\[+(2.25{{q}^{2}}-6.25{{q}^{2}})\] \[=4{{p}^{2}}-4{{q}^{2}}\]                 (vi) \[{{(ab+bc)}^{2}}-2a{{b}^{2}}c\]        \[{{(ab+bc)}^{2}}-2a{{b}^{2}}c\]\[=\{{{(ab)}^{2}}+2(ab)\,(bc)\]\[+{{(bc)}^{2}}\}\,-2a{{b}^{2}}c\]                 \[=({{a}^{2}}{{b}^{2}}+2a{{b}^{2}}c+{{b}^{2}}{{c}^{2}})\,-2a{{b}^{2}}c\] \[={{a}^{2}}{{b}^{2}}\,+(2a{{b}^{2}}c-2a{{b}^{2}}c)+{{b}^{2}}{{c}^{2}}\]                  |Combining the like terms \[={{a}^{2}}{{b}^{2}}+{{b}^{2}}{{c}^{2}}\]                 (vii) \[{{({{m}^{2}}-{{n}^{2}}m)}^{2}}\,+2{{m}^{3}}{{n}^{2}}\]      \[{{({{m}^{2}}-{{n}^{2}}m)}^{2}}\,+2{{m}^{3}}{{n}^{2}}\] \[=\{{{({{m}^{2}})}^{2}}-2({{m}^{2}})({{n}^{2}}m)\,+{{({{n}^{2}}m)}^{2}}\}+2{{m}^{3}}{{n}^{2}}\] \[=({{m}^{4}}-2{{n}^{2}}{{m}^{3}}+{{n}^{4}}{{m}^{2}})+2{{m}^{3}}{{n}^{2}}\] \[={{m}^{4}}+(2{{m}^{3}}{{n}^{2}}-2{{n}^{2}}{{m}^{3}})+{{n}^{4}}{{m}^{2}}\]                      |Combining the like terms                 \[={{m}^{4}}+{{n}^{4}}{{m}^{2}}\]