Bunuel wrote:

Which of the following is closest to the value of \((\sqrt{21}-\sqrt{7})^2\)?

A. 4

B. 5

C. 7

D. 16

E. 196

Foiling we have:

(√21)^2 + (√7)^2 - 2(√21)(√7) = 21 + 7 - 2√147

Since √147 is a little more than 12, we have:

21 + 7 - 2(12) = 28 - 24 = 4

Alternate Solution:

We can use approximation to solve this problem quickly. We see that √21 is close to 5, and √7 is close to 3. Therefore (5 - 3)^2 = 4.

Alternate Solution:

We note that √21 = √(7*3) = (√7)*(√3). Then,

(√21 - √7)^2 = (√7*√3 - √7)^2 = [(√7)^2][√3 - 1]^2 = 7*(√3 - 1)^2

We note that √3 is approximately 1.7; thus we can approximate as

(√21 - √7)^2 ≈ 7 * (1.7 - 1)^2

(√21 - √7)^2 ≈ 7 * (0.7)^2

(√21 - √7)^2 ≈ 7 * (0.49) = 3.43

The closest value among the choices is A.

Answer: A

_________________

Scott Woodbury-Stewart

Founder and CEO

GMAT Quant Self-Study Course

500+ lessons 3000+ practice problems 800+ HD solutions