NandishSS wrote:
HI
GMATGuruNY,
MentorTutoring GMATBustersCan you please help me with this question ?
Hello,
NandishSS, and thank you for mentioning me. This question took me about 55 seconds to answer (correctly, ha ha). A little number sense goes a long way. Keep in mind, no matter what root is being asked about, if the number under the root symbol is NOT 0 or 1, either of which would lead to the same answer (e.g., √0 = 0, or √1 = 1), and, of course, you are not dealing with complex numbers such as
i (which the GMAT™ does not have on it anyway), then the root
must be greater than 1. How do you know? Because, quite simply, looking at, say,
\(\sqrt[9]{3}\)
you could multiply 1 * 1 * 1 * 1 * 1 * 1 * 1 * 1 * 1 and you will never get the product to equal 3. It can only equal 1. Without doing the math, you can say with certainty that 1-something times itself 9 times would work up to 3. Likewise, you could interpret each item in the chain in such a manner:
\(\sqrt{10}\) = slightly more than 3, since √9 = 3;
\(\sqrt[3]{9}\) = slightly more than 2, since the cube root of 8 is 2;
\(\sqrt[4]{8}\) = more than 1, since, again, we know the
cube root, or third root, of 8 is 2;
...
There is no point in completing the rest, since the number under the root is greater than 1. Thus, the sum becomes
3-something + 2-something + (1-something) * 7
Even if we ignore the
-something, the sum must equal
at least3 + 2 + 1 * 7, which gives us 12
Thus, putting the
-something back into the mix, we know that the sum in the original question
must be greater than 12, and we saved ourselves a whole lot of trouble trying to calculate specifics.
I hope that helps. If you have further questions, feel free to ask.
- Andrew