Tricky OG 12 Question- need help

Anyone? Please? I really would like to understand why the OG does this.

Thank you,

When you have a root in the denominator, like 1/[√ x - √ y]. Then if you multiply numerator and denominator by the opposite sign of the denominator, then the root sign in the denominator will go by the formula.

1/[a-b] * [a + b]/[a + b] = a + b/ a^2 - b^2

I hope this is enough.

The goal when dealing with radicals is to square as many of the radicals as possible to make them "normal" (a.k.a. not under the square root sign).

So how can we make the denominator "normal" by getting rid of the radicals?
By squaring! Let's use the conjugate of the denominator

Think of it like a factoring problem --

Substitute w = sqrt(n+1) and j = sqrt(n)

Here's the denominator....
(w-j)(w+j) =(w^2 - j^2)=(n+1 - n) = 1


Bingo -- the denominator has been simplified.

Now both of the denominator's square roots have been squared and made "normal" again. Since we multiplied the denominator by w+j, we have to multiply the numerator by w+j, too.

As shown above, the denominator is equal to 1

we're left with w+j...

and that's choice E!


Whenever you have radicals in the denominator like this, try to multiply the denominator by its conjugate so the radicals square and cancel out.


Hope this helps!