pepo wrote:

(\sqrt{15 + 4 [square_root]14}[/square_root] + \sqrt{15 - 4 [square_root]14}[/square_root])^2

A) 28

B) 30

C) 32

D) 34

E) 36

How much is probable to get a question similar to this one on test day?

What level do you think is this question?

Thanks in advance

Dear

pepoI'm happy to respond.

If I understand correctly, this is the mathematical expression you were trying to convey:

Attachment:

root expression.JPG [ 10.43 KiB | Viewed 1212 times ]
Is this correct? It is not clear to me how to use the LaTex math notation to generate nested roots. Perhaps Bunuel or another expert can demonstrate that.

If I have the expression correct, this is a 100% valid question, a very clever question, very easy to do in one's head. It's important to know the fundamental polynomial patterns,

the Square of a Difference and the

Difference of Two Squares.

The square of the difference formula is

\((A - B)^2 = A^2 - 2AB + B^2\)

First look at the square of the individual terms.

\(A^2 = 15 + 4\sqrt{14}\)

\(B^2 = 15 - 4\sqrt{14}\)

When we add those, the radical expressions cancel, and we just get 30.

The cross term is more interesting. We combine the two radicals, and underneath the radicals, we get the difference of two squares pattern:

\((p + q)( p - q) = p^2 - q^2\)

Think about what's under the radical.

\((15 + 4\sqrt{14})(15 - 4\sqrt{14}) = 15^2 - (4^2)(14) = 15^2 - (16)(14)\)

We don't even have to do the calculation at the end: we simply need to recognize some

advanced math factoring. From the difference of two squares,

\((K + 1)(K - 1) = K^2 - 1\)

For any integer K, the product of the integer one bigger (K + 1) and one smaller (K - 1) will always be one less than K^2. Without doing a calculation, we know that (16)(14) is exactly one less than 15^2. Thus, everything under the radical simplifies to 1. Thus

\(2AB = 2\)

and

\((A - B)^2 = A^2 - 2AB + B^2 = 30 - 2 = 28\)

Answer =

(A)Yes, this problem is full of material you should know well by test day. Please let me know if you have any questions.

Mike

_________________

Mike McGarry

Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)