If  \(x=\sqrt10+3\sqrt9+4\sqrt8+5\sqrt7+6\sqrt6+7\sqrt5+8\sqrt4+9\sqrt3+10\sqrt2\) , then Which of the Following Must be True? GMAT Problem Solving

Question: If \(x=\sqrt10+3\sqrt9+4\sqrt8+5\sqrt7+6\sqrt6+7\sqrt5+8\sqrt4+9\sqrt3+10\sqrt2\) , then which of the following must be true?

1. x > 12
2. 10 < x < 12
3. 8 < x < 10
4. 6 < x < 8
5. x < 6

“If \(x=\sqrt10+3\sqrt9+4\sqrt8+5\sqrt7+6\sqrt6+7\sqrt5+8\sqrt4+9\sqrt3+10\sqrt2\) , then which of the following must be true?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. 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:

There is a little trick in the question: any positive integer root form a number more than 1 will be more than 1

For example: \({1000}\sqrt2>1\)

Now  \(\sqrt10\)> 3 (as \(3^2=9\) ) and \(3\sqrt9\) > 2 (as \(2^3=8\))

Thus, x = (Number more than 3) + (Number more than 2) + (7 numbers more than 1) = (Number more than 5 ) + (Number more than 7) = (Number more than 12)

Correct Answer: A

Approach Solution 2:

Consider the smallest value:

a = \(10\sqrt2\)

\(a^{10}=2\)

Since \(1^{10}\) = 1, the value of a must be greater than 1

Implication:

The 7 smallest values are all greater than 1, summing to a value greater than 7

Consider the two greatest values:

Since, \(\sqrt9=3,\sqrt10>3\)

Since, \(3\sqrt8=2,3\sqrt9>2\)

Thus:

x = (more than 3) + (more than 2) + (more than 7) = (more than 12)

Correct Answer: A

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