For questions like these, where you have to find out whether a number is a perfect square or a perfect cube, always work with identifying squares/cubes of numbers which end with ZERO. For example, \(10^2\) = 100, \(20^2\) = 200 and so on. Similarly, \(10^3\) = 1000, \(20^3\) = 8000, \(30^3\) = 27000 and so on.
Also, the unit digit of a perfect cube corresponds uniquely to the unit digit of its cube root.
A perfect cube ending with 8 will have its cube root ending with 2.
A perfect cube ending with 6 will have its cube root ending with 6.
A perfect cube ending with 2 will have its cube root ending with 8.
Therefore, in the options, we can say that if 39316 is a perfect cube, its cube root will have the unit digit as 6; similar conclusion can be drawn about 27006. 46658/32768, if they are perfect cubes, will have their cube root ending with 2. 64012, if it’s a perfect cube will be ending with 8.
If a number ends with 0, finding out its cube is the easiest thing to do. That is,
If x = k0, then \(x^3\) = (\(k^3\))000. For example,
\(10^3\) = 1000, \(20^3\) = 8000, \(30^3\) = 27000, \(40^3\) = 64000 and so on.
From the above examples, one thing is very clear – 27006 and 64012 are not perfect cubes. Answer options B and E can be eliminated straight away.
This also means that the cube root of the other three numbers lies between 30 and 40. But why? Because all these numbers are between 27000 and 64000.
Now, \(35^3\) will be a number above 42000. This is because \(35^2\) = 1225. Even if we ignore the 25, 1200 * 35 will give us 42000.
The only number that ends with 6, between 30 and 40, is 36. 36 is more than 35. Therefore, \(36^3\) cannot be 39316. So, 39316 cannot be a perfect cube.
The only number that ends with 2, between 30 and 40 is 32. 32 is less than 35. Therefore, \(32^3\) cannot be 46658 (since 46658 is more than \(35^3\); clearly, \(32^3\) cannot be greater than \(35^3\), right?).
Therefore, 32768 IS the perfect cube and it IS the cube of 32. You may observe here that 32768 is also closer to 27000, so clearly between 46658 and 32768, it’s only 32768 which can be the cube of 32.
As mentioned before, working with perfect cubes is not about memorizing a set of numbers. It’s about knowing certain cubes which are easy to know, and then using these numbers with sound logic to eliminate numbers which cannot be perfect cubes.
Hope this helps!
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Crackverbal Prep Team
www.crackverbal.com