Problem Solving

The expression resembles this form:
(a - b)^2
where
a = (7 + (29)^0.5)^0.5
b = (7 - (29)^0.5)^0.5

(a - b)^2
a^2 + b^2 - 2ab
a^2 = 7 + (29)^0.5
b^2 = 7 - (29)^0.5
2ab = 2*[(7 + (29)^0.5)(7 - (29)^0.5)]^0.5
= 2*[20]^0.5
= 4*[5]^0.5

So (a - b)^2
= 7 + 7 - 4[5]^0.5
= 14 - 4[5]^0.5

[spoiler](C)[/spoiler] is the answer

aneesh.kg wrote:The expression resembles this form:
(a - b)^2
where
a = (7 + (29)^0.5)
b = (7 - (29)^0.5)

(a - b)^2
a^2 + b^2 - 2ab
a^2 = 7 + (29)^0.5
b^2 = 7 - (29)^0.5
2ab = 2*(7 + (29)^0.5)(7 - (29)^0.5)
= 2*(49 - 29) = 40

So (a - b)^2
= 7 + 7 - 40
= - 26

[spoiler](A)[/spoiler] is the answer

Wrong! You missed the whole root!!!

Ah, Yes I did. Thanks.
Corrected the post above.

Corrected Solution:

The expression resembles this form:
(a - b)^2
where
a = (7 + (29)^0.5)^0.5
b = (7 - (29)^0.5)^0.5

(a - b)^2
a^2 + b^2 - 2ab
a^2 = 7 + (29)^0.5
b^2 = 7 - (29)^0.5
2ab = 2*[(7 + (29)^0.5)(7 - (29)^0.5)]^0.5
= 2*[20]^0.5
= 4*[5]^0.5

So (a - b)^2
= 7 + 7 - 4[5]^0.5
= 14 - 4[5]^0.5

[spoiler](C)[/spoiler] is the answer

What is the value of [√(7+√29) - √(7-√29)]²?

-26

2√29

14-4√5

14

14+4√5

The square of a value must be nonnegative.
Eliminate A.

Ballpark.
[√(7+√29) - √(7-√29)]²

≈ [√(7+6) - √(7-6)]² Since √25=5 and √36=6, √29 is between 5 and 6.

≈ (√13 - 1)²

≈ 3² Since √9=3 and √16=4, √13 is between 3 and 4.

≈ 9.

Since we overestimated the value of √13-1, the correct answer must be less than 9.

Only answer choice C works:
14-4√5 ≈ 14-8 ≈ 6.

The correct answer is C.

If we couldn't ballpark because the answer choices were closer in value, we could use the following approach.

Remembering that (a-b)² = a²-2ab+b², use substitution:
Let x=7 and y=√29.
Substituting x=7 and y=√29 into [√(7+√29) - √(7-√29)]², we get:

(√(x+y) - √(x-y))²

= (x+y) - 2√((x+y)(x-y)) + (x-y)

= 2x - 2√(x²-y²)

= 2*7 - 2√(7² - 29)

= 14 - 2√20

= 14 - 4√5.

The correct answer is C.