Rationalize the denominator and simplify: 4 /4+sqrt x

Answer 1

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator.

The conjugate of 4 + sqrt(x) is 4 - sqrt(x).

Therefore,
4 / (4 + sqrt(x))
= 4(4 - sqrt(x)) / (4 + sqrt(x))(4 - sqrt(x))
= (16 - 4sqrt(x)) / (16 - x)

So, the simplified expression is:
(16 - 4sqrt(x)) / (16 - x)