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If =√3−√2√3+√2and =√3+√2√3−√2, find the value of (a+b)3 and a2+b2+ 5ab |
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Answer» a=3–√+2–√3–√−2–√a=3+23−2 =(3–√+2–√)(3–√+2–√)(3–√−2–√)(3–√+2–√)=(3+2)(3+2)(3−2)(3+2) =(3–√+2–√)23−2=(3+2)23−2 =((3–√)2+(2–√)2+2(3–√)(2–√)=((3)2+(2)2+2(3)(2) =3+2+26–√=3+2+26 =5+26–√=5+26 b=3–√−2–√3–√+2–√b=3−23+2 =(3–√−2–√)(3–√−2–√)(3–√+2–√)(3–√−2–√)=(3−2)(3−2)(3+2)(3−2) =(3–√−2–√)23−2=(3−2)23−2 =((3–√)2+(2–√)2+−2(3–√)(2–√)=((3)2+(2)2+−2(3)(2) =3+2−26–√=3+2−26 =5−26–√=5−26 Now a2+b2a2+b2 We know that (x+y)2=x2+y2+2xy(x+y)2=x2+y2+2xy ⟹x2+y2=(x+y)2−2xy⟹x2+y2=(x+y)2−2xy If we take x=ax=a and y=by=b then a2+b2a2+b2 =(a+b)2−2ab=(a+b)2−2ab =(5+26–√+5−26–√)2−2((5+26–√)(5−26–√))=(5+26+5−26)2−2((5+26)(5−26)) =(10)2−2(25−24)=(10)2−2(25−24) Here we use IDENTITY (a+b)(a−b)=a2−b2(a+b)(a−b)=a2−b2 =100−2=100−2 =98=98 |
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