If a=1/7-4√3 and b=1/7+4√3, find a³+b³
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a = 1/( 7 -4√3) = ( 7 + 4√3)/( 7-4√3)(7+4√3)
= (7+ 4√3)/(7² -(4√3)²) = 7 + 4√3
similarly ,
b = 1/( 7 +4√3) = ( 7 - 4√3)
now ,
a + b = ( 7 + 4√3) + ( 7-4√3) = 14
also b = 1/a
hence,
a³ + b³ = a³ + 1/a³ = ( a + 1/a)³ -3a×1/a( a +1/a) = ( 14)³ - 3( 14 ) = 14 × ( 196- 3)
= 14 × 193 = 2702
= (7+ 4√3)/(7² -(4√3)²) = 7 + 4√3
similarly ,
b = 1/( 7 +4√3) = ( 7 - 4√3)
now ,
a + b = ( 7 + 4√3) + ( 7-4√3) = 14
also b = 1/a
hence,
a³ + b³ = a³ + 1/a³ = ( a + 1/a)³ -3a×1/a( a +1/a) = ( 14)³ - 3( 14 ) = 14 × ( 196- 3)
= 14 × 193 = 2702
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6
Answer:
67228 is the answer i think
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