| Simplify |
| (a) \[{{({{a}^{2}}-{{b}^{2}})}^{2}}\] |
| (b) \[{{(2x+5)}^{2}}-{{(2x-5)}^{2}}\] |
| (c) \[{{\left( 7m+8n \right)}^{2}}+{{\left( 7m+8n \right)}^{2}}\] |
| (d)\[{{\left( 4m+5n \right)}^{2}}+{{\left( 5m+4n \right)}^{2}}\] |
Answer:
(a) \[{{({{a}^{2}}{{b}^{2}})}^{2}}\] Use the identity, \[{{(ab)}^{2}}={{a}^{2}}2ab+{{b}^{2}}\] \[={{({{a}^{2}})}^{2}}2{{a}^{2}}{{b}^{2}}+\text{ }{{({{b}^{2}})}^{2}}\] \[={{a}^{4}}2{{a}^{2}}{{b}^{2}}+{{b}^{4}}\] (b) \[{{(2x+\text{ }5)}^{2}}{{(2x5)}^{2}}\] Use the identities, \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\] \[{{(ab)}^{2}}={{a}^{2}}2ab+{{b}^{2}}\] \[=\{{{(2x)}^{2}}+2(2x)\left( 5 \right)+{{\left( 5 \right)}^{2}}\}{{(2x5)}^{2}}\] \[=\{4{{x}^{2}}+4x\times 5+25\}\{{{(2x)}^{2}}2(2x)\left( 5 \right)+{{\left( 5 \right)}^{2}}\] \[=(4{{x}^{2}}+20x+25(4{{x}^{2}}20x+25)\] \[=\text{ }4{{x}^{2}}+20x+254{{x}^{2}}+20x25\] \[=\text{ }20x+\text{ }20x\] \[=\text{ }40x\] (c) \[{{(7m8n)}^{2}}+\text{ }{{(7m+\text{ }8n)}^{2}}=\{{{(7m)}^{2}}2(7m)\] \[(8n)+{{(8n)}^{2}}\}+\{{{(7m)}^{2}}+2(7m)\text{ }(8n)+{{(8n)}^{2}}\}\] \[=\text{ }49{{m}^{2}}112mn+64{{n}^{2}}+\text{ }49{{m}^{2}}+\text{ }112mn+64{{n}^{2}}\] \[=98{{m}^{2}}+128{{n}^{2}}\] (d) \[{{(4m+5n)}^{2}}+{{(5m+4n)}^{2}}\] Use the identity, \[{{(a+b)}^{2}}={{a}^{2}}+\text{ }2ab+{{b}^{2}}\] \[=\{{{(4m)}^{2}}+2(4m)\text{ }(5n)+{{(5n)}^{2}}\}+\{{{(5m)}^{2}}+2\text{ }(5m)(4n)+{{(4n)}^{2}}\}\]\[=(16{{m}^{2}}+\text{ }40mn+\text{ }25{{n}^{2}})+\text{ }(25{{m}^{2}}+\text{ }40mn+\text{ }16{{n}^{2}})\]\[=\left( 16+25 \right){{m}^{2}}+\left( 40\text{ }+\text{ }40 \right)mn+\text{ }125\text{ }+\text{ }16){{n}^{2}}\] \[=\text{ }41{{m}^{2}}+\text{ }80mn+\text{ }41{{n}^{2}}\]
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