Obtain the binomial expansion of ( 2 - square root of 3) to the power of 6 in the form - Maths - Binomial Theorem

Question

Obtain the binomial expansion of ( 2 - square root of 3) to the
power of 6 in the form a + b square root of 3, where a and b are
integers State the corresponding result for the expansion ( 2 +
square root of 3) to the power of 6 and show that ( 2 - square root
of 3) to the power of 6 is the reciprocal of ( 2 + square root of
3) to the power of 6?

Ajanta Trivedi · Accepted Answer

the given binomial expression is:
2-36=C06 (2)6+C16 (2)5 (-3)+C26 (2)4 (-3)2+C36 (2)3
(-3)3+                       C46
(2)2 (-3)4+C56 (2) (-3)5+C66
(-3)6(2-3)6=1*64-6*323+6*52*16*3-6*5*43*2*8*33+6*52*4*9                      -6*2*93+1*27(2-3)6=64-1923+720-4803+540-1083+27=1351-7803since (2-3)6=a+b3therefore by comparing we have:a=1351 and b=-780
and similarly (2+3)6=1351+7803
2+3=(2+3)*2-32-3=4-32-3=12-3therefore 2+3=12-3and (2+3)6=1(2-3)6

hope this helps you