\({\sqrt{9+\sqrt{80}} + \sqrt{9-\sqrt{80}}})^2\) GMAT Problem Solving

Question: (\(\sqrt{9+\sqrt{80}}\) + \(\sqrt{9-\sqrt{80}}\))²

1. 1
2. 9 - 4*\(\sqrt{5}\)
3. 18 - 4*\(\sqrt{5}\)
4. 18
5. 20

“(\(\sqrt{9+\sqrt{80}}\) + \(\sqrt{9-\sqrt{80}}\))²”​- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

This is a square root problem. This can be solved using the following formulas given below:

1. \((x+y)^2\) = \(x^2 \)+ 2xy + \(y^2\)

\((x-y)^2\) = \(x^2 \)- 2xy + \(y^2\)

2. (x + y)(x - y) = \(x^2 \)- \(y^2\)

So, this (\(\sqrt{9+\sqrt{80}}\)+ \(\sqrt{9-\sqrt{80}}\))² can be further solved:
(\(\sqrt{9+\sqrt{80}}\) +\(\sqrt{9-\sqrt{80}}\))² = (\(\sqrt{9+\sqrt{80}}\))² + 2 (\(\sqrt{9+\sqrt{80}}\) + \(\sqrt{9-\sqrt{80}}\)) + ( \(\sqrt{9-\sqrt{80}}\))²
= 9 +\(\sqrt{80}\)+ 2 \(\sqrt{(9+\sqrt{80})+(9-\sqrt{80})}\)+ 9 - \(\sqrt{80}\)
= 18 + 2 \(\sqrt{9^2-(\sqrt{80})^2}\)
= 18 + 2 \(\sqrt{81-80}\)
= 18 + 2
= 20
Hence the value of (\(\sqrt{9+\sqrt{80}}\) + \(\sqrt{9-\sqrt{80}}\))² is 20.

Correct Answer: E

Approach Solution 2:

Let,
h A = \(\sqrt{9+\sqrt{80}}\)
B = \(\sqrt{9-\sqrt{80}}\)
We can use the\( (A+B)^2\) formula to solve the equation.
\( (A+B)^2\) = \(A^2\)+ 2AB + \(B^2\)
Putting the values of A and B, we get,
\(A^2\)= (\(\sqrt{9+\sqrt{80}}\))² = 9 +\(\sqrt{80}\)
\(B^2\)= ( \(\sqrt{9-\sqrt{80}}\))² = = 9 - \(\sqrt{80}\)
AB = \(\sqrt{9+\sqrt{80}}\) * \(\sqrt{9-\sqrt{80}}\)
=\(\sqrt{(9+\sqrt{80})+(9-\sqrt{80})}\)= \(\sqrt{1}\)= 1
Solving (9 + \(\sqrt{80}\))(9-\(\sqrt{80}\))
= \(9^2\)- (\(\sqrt{80}\)\()^2\)
= 81 - 80
= 1
Hence , (\(\sqrt{9+\sqrt{80}}\) + \(\sqrt{9-\sqrt{80}}\))²
= (9 + \(\sqrt{80}\)) + 2*1 +(9-\(\sqrt{80}\))
=20
Hence the value of (\(\sqrt{9+\sqrt{80}}\) + \(\sqrt{9-\sqrt{80}}\))² is 20.

Correct Answer: E

Approach Solution 3:

Notice that the expression is in the form (x + y)², where x = √(9 + √80) and y = √(9 - √80)

We know that (x + y)² = x² + 2xy + y²

If x = √(9 + √80), then x² = 9 + √80
If y = √(9 - √80), then y² = 9 - √80
Finally, xy = [√(9 + √80)][√(9 - √80)] = 81 - 80 = 1

So, we get:
(x + y)² = x² + 2xy + y²
= (9 + √80) + 2(1) + (9 - √80)
= 9 + √80 + 2 + 9 - √80
= 20

Correct Answer: E

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