• # question_answer 21)                 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

(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                 $={{m}^{4}}+{{n}^{4}}{{m}^{2}}$