What is the value of 2/(2 - sqrt2)?

[BTGModeratorVI]

What is the value of 2/(2−√2)?

(A) 1
(B) 2
(C) 1 + √2
(D) 2 + √2
(E) 4\
Answer: D
Source: Veritas Prep

[Brent@GMATPrepNow]

What is the value of 2/(2−√2)?

(A) 1
(B) 2
(C) 1 + √2
(D) 2 + √2
(E) 4\
Answer: D
Source: Veritas Prep

Given: 2/(2 - √2)

Our goal is to rationalize ("fix") the denominator so that it doesn't have a root.
We'll do this by multiplying the numerator and denominator by the conjugate of 2 - √2, which is 2 + √2

Numerator: 2(2 + √2) = 4 + 2√2
Denominator: (2 - √2)(2 + √2) = 4 - 2 = 2

So, 2/(2 - √2) = (4 + 2√2)/2 = 2 + √2

Answer: D

Cheers,
Brent

[Brent@GMATPrepNow]

What is the value of 2/(2−√2)?

(A) 1
(B) 2
(C) 1 + √2
(D) 2 + √2
(E) 4\
Answer: D
Source: Veritas Prep

Another approach:

All GMAT test-takers should all memorize the following approximations:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2

So, 2/(2 - √2) ≈ 2/(2 - 1.4)
≈ 2/(0.6)
≈ 3.3

Now check the answer choices...
(A) 1 nope (ELIMINATE)
(B) 2 nope (ELIMINATE)
(C) 1 + √2 ≈ 1 + 1.4 ≈ 2.4 nope (ELIMINATE)
(D) 2 + √2 ≈ 2 + 1.4 ≈ 3.4 looks good. KEEP
(E) 4 nope (ELIMINATE)

Answer: D

Cheers,
Brent

[Scott@TargetTestPrep]

What is the value of 2/(2−√2)?

(A) 1
(B) 2
(C) 1 + √2
(D) 2 + √2
(E) 4\
Answer: D
Source: Veritas Prep

When we have a radical or radical expression in the denominator, we must rationalize the denominator, in this case by multiplying the numerator and denominator by the conjugate of (2 - √2), which is (2 + √2).

2 / (2−√2) x (2+√2) / (2+√2)

Note that when the expressions in the denominator are FOILed, we obtain 4 + 2√2 - 2√2 - 2, which simplifies to 4 - 2.

2(2+√2) / (4 - 2)

2(2+√2) / 2

2 + √2

Answer: D