Bunuel wrote:

JackSparr0w wrote:

Bunuel wrote:

Official Solution:

The expression \(7 + \sqrt{48}\) can be rewritten as \(4 + 4\sqrt{3} + 3\) which is the same as \((2 + \sqrt{3})^2\). Therefore,\(\sqrt{7 + \sqrt{48}} - \sqrt{3} = 2 + \sqrt{3} - \sqrt{3} = 2\).

Answer: C

Is there anything specific here that we should pick up on, that would tell us that we should break up the problem this way?

the √48 can be broken down. I get that, and it seems intuitive enough. But is there a way to realize that we need to break the 7 into 4+3, and then use foil?

I guess Im just trying to figure out what the "real time" thought process should be when one sees this.

Thanks

I guess such kind of tricks should come with practice.

I came up with a generalized approach.

Visually superimpose the following form onto your expression (or sub-expression):

\(a±b\sqrt{c}\)

Then test to see if

\(\sqrt{a-c} = \frac{b}{2}\)

If so then you can rearrange your initial expression as

\((a-c) ± b\sqrt{c} + c\)

Try this with the problem example, where a = 7; b = 4; and c = 3

Also see with another example here:

\(14 + 6\sqrt{5}\)

a = 14

b = 6

c = 5

Test it:

\(\sqrt{14-5} = \frac{6}{2}\)

3 = 3

It passes the test, so we can rewrite the original expression as:

\(9 + 6\sqrt{5} + 5\)

Which can then be factored into

\((3+\sqrt{5})^2\)

I'm not sure if there are enough occurrences of this to justify memorizing this trick, but it's a tool you can use nonetheless.

I personally like Magsy's method.