Question

What is the value of (2+root5)^1/3 + (2-root5)^1/3 ?

Answer 1

The value is `-2`

Explanation:

Let `x= (2+sqrt5)^(1/3)+(2-sqrt5)^(1/3)` then

`x^3= {(2+sqrt5)^(1/3)+(2-sqrt5)^(1/3)}^3`

Reminder:

[`(a+b)^3=a^3+b^3+3ab(a+b) , a^2-b^2=(a+b)(a-b)`]

and let `a=(2+sqrt5), b=(2-sqrt5):. ab=4-5=-1`

`:.x^3=(2+sqrt5)^(3*1/3)+(2-sqrt5)^(3*1/3)+3(2+sqrt5)(2-sqrt5)(2+sqrt5+2-sqrt5)`

or `x^3=2+cancel(sqrt5)+2-cancel(sqrt5)+3(4-5)(2+cancel(sqrt5)+2-cancel(sqrt5))`

or `x^3=4+3(-1)(4) or x^3 =4 -12 or x^3= -8` or

`x = (-8)^(1/3) =-2` . The value is `-2` [Ans]