Question

Properties of perfect square and perfect cube

Answer 1

Cubes

The cube of a number is the number raised to the power 3. Thus, 
cube of 2 = 23 = 2 x 2 x 2 = 8, 
cube of 5 = 23 = 5 x 5 x 5 = 125.

Perfect Cube


We know that 23 = 8, 33 = 27, 263 = 216, 73 = 343, 103= 1000.
The numbers 8, 27, 216, 343, 1000, … are called perfect cubes. A natural number is said to be a perfect cube, if it is the cube of some natural number, i.e.,
A natural number n is a perfect cube if there exists a natural number m such that  m X m X m = n.

Procedure :

Step I- Obtain the natural number.

Step II- Express the given natural number as a product of prime factors.

Step III- Group the factors in triples in such a way that all the three factors in each triple are equal.

Step IV- If no factor is left over in grouping in step III, then the number is a perfect cube, otherwise not

Answer 2

Properties of a perfect square...
1. Numbers ending in 2, 3, 7 or 8 is never a perfect square but on the other hand, all the numbers ending in 1, 4, 5, 6, 9, 0 are not square numbers.
2.A number ending in an odd number of zeros is never a perfect square.
3.The square of an even number is always even.
4. The square of an odd number is always odd. 
5. The square of a proper fraction is smaller than the fraction. 

Properties of a perfect cube...
1. Cubes of all odd natural numbers are odd.
2. Cubes of all even natural numbers are even.
3. The cube of a negative integer is always negative.
4. For any rational number ab, we have (a/b)3=(a3/b3).
6. Cubes of the numbers ending in digits 1, 4, 5, 6 and 9 are the numbers ending in the same digit. Cubes of numbers ending in digit 2 ends in digit 8 and the cube of numbers ending in digit 8 ends in digit 2. The cubes of the numbers ending in digits 3 and 7 ends in digit 7 and 3 respectively.



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