In the expression (sqrt(x+13) - 3)/(x + 4), what is the simplest form after rationalizing the numerator?

To simplify the expression ((\sqrt{x + 13} - 3)/(x + 4)) by rationalizing the numerator, the goal is to eliminate the square root in the numerator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the numerator, which is ((\sqrt{x + 13} + 3)).

When you multiply the numerator by its conjugate, you apply the difference of squares formula:

[

(\sqrt{x + 13} - 3)(\sqrt{x + 13} + 3) = (\sqrt{x + 13})^2 - 3^2 = (x + 13) - 9 = x + 4

]

So after multiplying, the numerator becomes (x + 4). The denominator, after multiplying by ((\sqrt{x + 13} + 3)), becomes:

[

(x + 4)(\sqrt{x + 13} + 3)

]

Thus, we can rewrite the entire expression:

[

\frac{x + 4}{(x + 4)(\sqrt{x + 13} + 3)}

]

At this stage, you can