What is the Value of \([\sqrt{7+\sqrt{29}} - \sqrt{7-\sqrt{29}}]^2\) GMAT Problem Solving

Question: What is the value of \( [\sqrt{7+\sqrt{29}} - \sqrt{7-\sqrt{29}}]^2\)

1. -26
2. \(2\sqrt{29}\)
3. 14 - \(4\sqrt{5}\)
4. 14
5. 14 + \(4\sqrt{5}\)

“What is the value of \( [\sqrt{7+\sqrt{29}} - \sqrt{7-\sqrt{29}}]^2\)“- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “ GMAT Prep”. 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:

It is asked What is the value of \( [\sqrt{7+\sqrt{29}} - \sqrt{7-\sqrt{29}}]^2\)

One can apply two properties:
\((a – b)^2 = a^2 – 2ab + b^2\)
\((a + b) (a – b) = a^2-b^2\)

\( [\sqrt{7+\sqrt{29}} - \sqrt{7-\sqrt{29}}]^2\)= \([\sqrt{7-\sqrt{29}}]^2\)- \(2[\sqrt{ [ 7 +\sqrt{29} ] [ 7 - \sqrt{29} ]}\) + \([\sqrt{7-\sqrt{29}}]^2\)

= \(7 + \sqrt{29} - 2 \sqrt{49-29}+ 7 -\sqrt{29}\)

= \(14 - 2\sqrt{20}\)

=\( 14 - 4\sqrt{5}\)

The answer is \( 14 - 4\sqrt{5}\).

Correct Answer: C

Approach Solution 2:

There is another approach to solve the question.
If time is of the essence, you can also employ estimates. There are enough options when examining the response possibilities for an estimate.
We know

\(5^2=25\)
\(6^2= 36\)

So using estimate the\(\sqrt{29}\)to be 5.5.
Now 7 + 5.5 = 12.5 and 7 -5.5 = 1.5.

\(3^2= 9\) and \(4^2= 16\)

therefore calculate \(\sqrt{12.5}\) to be 3.5 and\(\sqrt{5}\) to be little more than 1, or just assume 1, as in the example.

Now you have

\([3.5-1]^2= 6.xx\)

When you look at the possible answers, you can quickly cross off options A (too small), D, and E. (too big).
Our options now are between B and C.
If we observe that B has \(\sqrt{29}\), you already knew it was 5.5.
It can be multiplied by two to get a result bigger than 10.
The fact that you have 14 - 4*2.something makes Answer C look good.
This number will be around 6.
Therefore the Answer is C

The answer is \( 14 - 4\sqrt{5}\).

Correct Answer: C

Approach Solution 3:

Looking at the answer choices, they are spread out far enough to estimate.

5^2 = 25 and 6^2 = 36. So estimate the sqrt29 to be 5.5.

Now 7 + 5.5 = 12.5 and 7 -5.5 = 1.5.
3^2 = 9 and 4^2 = 16. So estimate the sqrt12.5 to be 3.5 and Sqrt1.5 to be a little more than 1, or just assume 1.

Not you have (3.5 -1)^2 = 6.something

Looking at the answer choices, you can immediately cross of A (too small), D and E (too big).
Now you are left between b and C. If you notice b has sqrt 29, which you already assumed to be 5.5. Multiplying it by 2 will give you a number greater than 10. Answer C looks good because you have 14 - 4*2.something. This value will be closer to 6

The answer is \( 14 - 4\sqrt{5}\).

Correct Answer: C

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