If x=3+2√2,find the value of( x to the power 3)-1/(x to the power 3).And plz mention why (4√2) to the power 3 =128√2?

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Answer 1 · 2018-03-26T10:57:40

# 140sqrt2#.

We are given that, #x=3+2sqrt2#.

#:. 1/x=1/(3+2sqrt2)=1/(3+2sqrt2)xx1#,

#=1/(3+2sqrt2)xx(3-2sqrt2)/(3-2sqrt2)#,

#=(3-2sqrt2)/{3^2-(2sqrt2)^2}=(3-2sqrt2)/(9-8)#.

# rArr 1/x=3-2sqrt2#.

Hence, #x^2+1/x^2=(3+2sqrt2)^2+(3-2sqrt2)^2#,

#=2{3^2+(2sqrt2)^2}#.

# rArr x^2+1/x^2=34#.

Thus, we have, #x-1/x=4sqrt2, and, x^2+1/x^2=34#.

Finally, #x^3-1/x^3=(x-1/x)(x^2+1+1/x^2)#,

#=(4sqrt2)(34+1)=140sqrt2#.

Answer 2 · 2018-03-26T11:11:57

# 140sqrt2#.

Another Method to solve the Problem :

#x=3+2sqrt2#.

#:. 1/x=1/(3+2sqrt2)=1/(3+2sqrt2)xx(3-2sqrt2)/(3-2sqrt2)#.

#:. 1/x=(3-2sqrt2)#.

#:. x-1/x=(3+2sqrt2)-(3-2sqrt2)=4sqrt2#.

Cubing, #(x-1/x)^3=(4sqrt2)^3=4^3*(sqrt2)^3#.

#:. x^3-1/x^3-3*x*1/x*(x-1/x)=64*(color(red)(sqrt2*sqrt2)*sqrt2)#.

#:. x^3-1/x^3-3(x-1/x)=64*(color(red)2sqrt2)=128sqrt2#.

#:. x^3-1/x^3-3(4sqrt2)=128sqrt2#.

# rArr x^3-1/x^3=128sqrt2+12sqrt2=140sqrt2#, as before!