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If x=(√2+1)to the power -1/3 prove that (x-x to the power -1) ³ +3(x-x to the power -1) + 2=0

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by sheerin

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by sheerin

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To prove (x-x^ -1) ³ +3(x-x ^ -1) + 2=0

LHS=(x-1/x) ³ +3(x-1/x ) + 2=x³-3(x)(1/x)(x-1/x)-1/x³+3(x-1/x )+2

=x³-3(x-1/x)-1/x³+3(x-1/x )+2

=x³-3(x-1/x)+3(x-1/x )-1/x³+2

=x³-1/x³+2

x³={(√2+1)^( −1/3)}³=(√2+1)^( −1) =1/(√2+1) =1(√2−1)/{(√2+1)(√2−1)}

=1(√2−1)/{2−1} =(√2−1)

and 1/x³=(√2+1)

Hence x³-1/x³+2 =(√2−1)−(√2+1)+2

=√2−1−√2−1+2 =−2+2=0=RHS Hence proved