Equivalent Expressions

[turophile]

I have an exercise in my textbook that I solved with an expression of this form: [sqrt(a + b) - sqrt(a)] / b

The answer in my textbook is in this form: 1 / [sqrt(a + b) + sqrt(a)]

I'm know my answer agrees with one in the textbook, because they evaluate to the same thing. For example, if a = 9 and b = 16, they both evaluate to 1/8. However, I'm having trouble doing the algebra to get from my expression to the one in the textbook.

 

[galactus]

Multiply top and bottom by the conjugate.

\(\displaystyle \frac{(\sqrt{a+b}-\sqrt{a})}{b}\cdot \frac{(\sqrt{a+b}+\sqrt{a})}{(\sqrt{a+b}+\sqrt{a})}\)

The top becomes b. The b in the bottom cancels with the one in the top and we get:

\(\displaystyle \frac{1}{\sqrt{a+b}+\sqrt{a}}\)