simplifying radicals and fractional numbers

Question

$$\frac{a}{a-\sqrt{a^2-16}}$$ is the expression. I am not sure if i answered it right but please help me do this.

Answer 1

What you have is an expression, but not an equation. We can indeed simplify the expression.

We can multiply by the conjugate of the denominator, to obtain a difference of squares., e.g. $$\begin{align}\frac{a}{a-\sqrt{a^2-16}} & = \frac{a(a + \sqrt{a^2 - 16)}}{(a - \sqrt {a^2 - 16})(a + \sqrt{a^2 - 16})} \\ \\ & = \frac{a(a + \sqrt{a^2 - 16)}}{a^2 - (a^2 - 16)}\\ \\ & = \frac{{a(a + \sqrt{a^2 - 16)}}}{ 16}\end{align}$$

Added: What we've accomplished is called "rationalizing the denominator", which simply means getting the "radical" out of the denominator.