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?

2 Answers
Mar 26, 2018

140sqrt2.

Explanation:

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.

Mar 26, 2018

140sqrt2.

Explanation:

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!