Please help me simplify this:
y = x/(4-sqrt(16+x))
Rationalize the denominator. Multiply by the conjugate, or (4+sqrt (16+x)) to the numerator, and the denominator. This should simplify it.
To simplify the expression y = x/(4 - sqrt(16 + x)), we need to rationalize the denominator.
Step 1: Multiply both the numerator and denominator by the conjugate of the denominator, which is (4 + sqrt(16 + x)).
By doing this, we eliminate the square root in the denominator and make the expression easier to simplify.
y = [x * (4 + sqrt(16 + x))] / [(4 - sqrt(16 + x)) * (4 + sqrt(16 + x))]
Step 2: Simplify the denominator using the difference of squares formula.
The difference of squares formula states that (a + b) * (a - b) = a^2 - b^2.
In this case, the terms (4 + sqrt(16 + x)) and (4 - sqrt(16 + x)) are conjugates, so their product will give us the difference of squares.
(y = [x * (4 + sqrt(16 + x))] / [(4)^2 - (sqrt(16 + x))^2])
(y = [x * (4 + sqrt(16 + x))] / [16 - (16 + x)])
(y = [x * (4 + sqrt(16 + x))] / [16 - 16 - x])
(y = [x * (4 + sqrt(16 + x))] / [-x])
Step 3: Simplify the expression further by canceling out the x terms.
(y = -[4 + sqrt(16 + x)])
Therefore, the simplified expression for y is -[4 + sqrt(16 + x)].