Question

If `x = 7 + 4sqrt(3)`, find the value of `x + 1/x`.

Answer 1

Given: `x = 7 + 4sqrt(3)`

Step-wise calculation:

1. First, calculate `1/x`:

`1/x = 1/(7 + 4sqrt(3))`

Rationalize the denominator:

`1/(7 xx 4sqrt(3)) xx (7 - 4sqrt(3))/(7 - 4sqrt(3)) = (7 - 4sqrt(3))/((7)^2 - (4sqrt(3))^2`

`1/(7 xx 4sqrt(3)) xx (7 - 4sqrt(3))/(7 - 4sqrt(3)) = (7 - 4sqrt(3))/(49 - 48)`

`1/(7 xx 4sqrt(3)) xx (7 - 4sqrt(3))/(7 - 4sqrt(3)) = 7 - 4sqrt(3)`

2. Then, find `x + 1/x`:

`x + 1/x = (7 + 4sqrt(3)) + (7 - 4sqrt(3))`

`x + 1/x = 7 + 7 + 4sqrt(3) - 4sqrt(3)`

`x + 1/x = 14`