Question

If x =
$\sqrt[3]{2 + \sqrt[2]{3} } \: then \: {x}^{3 +} \: 1 \div {x}^{3 \: } \: =$
​

Answer 1

Answer :

x³ + 1/x³ = 4

Solution :

• Given : x = (2 + √3)^⅓
• To find : x³ + 1/x³ = ?

We have ,

=> x = (2 + √3)^⅓

=> x³ = 2 + √3

Thus ,

1/x³ = 1/(2 + √3)

Now ,

Rationalising the denominator of the term in RHS , we get ;

=> 1/x³ = (2 - √3)/(2 + √3)(2 - √3)

=> 1/x³ = (2 - √3)/[2² - (√3)²]

=> 1/x³ = (2 - √3)/(4 - 3)

=> 1/x³ = (2 - √3)/1

=> 1/x³ = 2 - √3

Now ,

=> x³ + 1/x³ = 2 + √3 + 2 - √3

=> x³ + 1/x³ = 4

Hence ,

x³ + 1/x³ = 4