Question

if (x+1/x)^2=3 find the value of x^3+1/x^3​

Answer 1

Step-by-step explanation:

Given:-

(x+1/x)^2=3

To find:-

(x+1/x)^2=3 find the value of x^3+1/x^3 ?

Solution:-

Given that

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

This is in the form of (a+b)^2

where a = x and b = 1/x

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

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

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

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

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

and from (1)

x+(1/x)=√3------------(3)

We know that

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

a^3+b^3 = (a+b)^3-3ab(a+b)

Now,

x^3+(1/x^3)

=>[x+(1/x)]^3-3(x)(1/x)[x+(1/x)]

=>[x+(1/x)]^3-3(1)[x+(1/x)]

=>[x+(1/x)]^3-3[x+(1/x)]

=>(√3)^3-3(√3)

=>(√3×√3×√3)-(3√3)

=>(3√3)-(3√3)

=>0

x^3+(1/x^3) = 0

Answer:-

The value of x^3+(1/x^3) for the given problem is 0

Used formulae:-

• (a+b)^2 = a^2+2ab+b^2

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