1.

X +1/x=3 then find x⁵+1/x⁵=? Please give full explanation.

Answer»

Step-by-step EXPLANATION:

Given:-

X +1/x=3

To find:-

find x⁵+1/x⁵

Solution:-

Given that

x + 1/x =3 -----------(1)

on squaring both sides then

(x + 1/x )^2=3^2

=>x^2 + 2(x)(1/x) + (1/x)^2 =9

(since (a+b)^2=a^2+2ab+b^2)

=>x^2 + 2 + (1/x)^2 =9

=>x^2 + (1/x)^2 = 9-2

x^2 + (1/x)^2 = 7 -------(2)

On cubing both side of the equation (1) then

(x + 1/x)^3=3^3

(since (a+b)^3 = a^3+ b^3 +3AB(a+b))

=>x^3 +1/x^3+ 3(x)(1/x)(x+1/x)=27

=>x^3+1/x^3+3(3)=27 (from(1))

=>x^3+1/x^3+9=27

=>x^3+1/x^3=27-9

=>x^3+1/x^3=18-------(3)

Now multiplying (2)&(3)

(x^2+ 1/x^2)(x^3+1/x^3)=7×18

=>(x^2×x^3)+(x^2/x^3)+(x^3/x^2)+(1/x^2×1/x^3)=126

=>x^5+ 1/x +x + 1/x^5=126

=>x^5 + 1/x^5 + 3=126

=>x^5+1/x^5=126-3

x^5+1/x^5=123

Answer:-

The value of x^5+1/x^5=123

Used formulae:-

  • (a+b)^2=a^2+2ab+b^2
  • (a+b)^3=a^3+b^3+3ab(a+b)


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