GMAT Quantitative Section. Урок 50. Arithmetic square root

В некоторых задачах на тесте GMAT в математической части вы столкнетесь с необходимостью работы с арифметическим квадратным корнем . Для успешного решения таких задач нужно обладать определенными навыками. Именно им и посвящен сегодняшний урок. Давайте рассмотрим теоритечисекую основу данного понятия, подкрепив практическими заданиями.

Today we will have a look at the most common tasks that contain arithmetic square root and learn, how to solve them in the shortest possible period of time.

(символ V обозначает квадратный корень - прим. редактора)

At first let’s look at the most simple tasks, in which you have only to count square root of some expression.

V(16*20 + 8*32) = ?

(A) 4V20
(B) 22  24
(C) 25
(D) 4V20 + 8V2
(E) 32

Since 16*20 + 8*32 = 576 = 24²,  V(16*20 + 8*32) = 24. The answer is B.

The second variant of solving this problem is to split off expression inside V: 16*20 + 8*32 = 16*(20 + 16) = 16*36 = 42*62 = (4*6)², the answer is the same: 24.

Sometimes it is useful to remember algebraic expressions and formulas and use them to solve problems with roots.

Following facts should be useful:

a² – b² = (a + b)(a - b); (a + b)² = a² + 2ab + b²; (ab)² = a²b²; a(b + c) = ab + ac.

Sample problems:
 (1 - V5)(1 + V5) = ?

(A) -4
(B) 2
(C) 6
(D) -4 - 2V5
(E) 6 - 2V5

Obviously, (1 - V5)(1 + V5) = 1² – (V5)² =1 – 5 = -4. A is the answer.

V(24+5V23)+V(24-5V23)=?

(A) 48
(B) V24
(C) 1
(D) 5V2
(E) 24-25V23

In order to rid the expression of square roots, let's first square the entire expression. We are allowed to do this as long as we remember to "unsquare" whatever solution we get at that end.

V(24+5V23)+V(24-5V23) -> (V(24+5V23)+V(24-5V23))²

Notice that the new expression is of the form (x+y)² where x=V(24+5V23) and y=V(24-5V23).

Recall that (x+y)²=x²+y²+2xy. This is one of the GMAT's favorite expressions.

Returning to our expression:
x²=24+5V23, while y²=24-5V23 and 2xy=2(24+5V23)(24-5V23).

Notice that x²+y² neatly simplifies to 48. This leaves only the 2xy expression left to simplify.

In order to simplify 2(24+5V23)(24-5V23), recall that (Va)(Vb)=V(ab).
Thus, 2(24+5V23)(24-5V23)=2(V((24+5V23)*(24-5V23)).

Notice that the expression under the square root sign is of the form (x-y)(x+y). And recall that (x+y)(x-y)=x²-y². This is another one of the GMAT's favorite expressions.

Returning to our expression:
2(V((24+5V23)*(24-5V23))= 2V(24²-(5V23)²)=2V(24²-25*23)=2V(576-575)=2V1=2.

Finally then: x²+y²+2xy=48+2=50.

But now we must remember to "unsquare" (or take the square root of) our answer: V50=V(25*2)=5V2.

Therefore, the correct answer is D.

Resume:
To solve problems with roots it is useful to remember algebraic expressions and formulas. Using them in time will help you to make hard problem much easier.

Material prepared by Ksenia Zueva,
GMAT consultant at MBA Strategy

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