One of the fundamental operations in mathematics is **multiplication**. Multiplication can be operated either between different numbers or the same number can be multiplied by itself. Still, when we multiply a number by itself n times, we name that operation differently: square or cube, depending on the value of n.

Let’s take a better look at squares and cubes of numbers and sort this multiplication problem out.

## Squares Of Numbers

The symbol of the square of a number is “2”. The square of the number 3 would be indicated by: 32 and the mathematical operation would look like this: 3 x 3 = 9, because we multiply the number by itself once, but the same number is used in the calculus 2 times.

Multiplying numbers by themselves bares the name of square because these particular numbers can be used to fill a perfect square. Square numbers will always form a filled-up square shape.

Let’s test this with some coins!

Pick up some coins and lay them on the desk. Try to set the coins according to the next square numbers:

32 = 3 x 3 , we would have 3 lines of coins, both horizontally and vertically, in total 9 coins.

What about the following squares: 22 , 42, 52 , 72 , 82 do their results form filled-shaped squares? Remember that the **square** must be filled with coins, just like in the image below:

All the numbers can be multiplied by themselves *n, *respectively 2 times, but only those which form solid shapes of squares are called **perfect squares**. You can check the lists of perfect square numbers **online**.

The **perfect square formula** is N = X2

### Did You Know That...?

- Perfect square numbers always end up with 0,1,4,5,6 or 9.
- A square number will always contain an even number of zeros;
- Numbers that contain 1 or 9 as the last digit will always have digit 1 at the end of its square number;
- Numbers that have 4 or 6 as the last digit will have a square number ending with 6;
- The square root of a perfect square number is always a whole number;
- The square number of odd numbers will always be odd, and the square number of an even number will always be even.

## Cubes Of Numbers

The symbol of the cube numbers is “3”. The cube of number 3 would be noted like: “33” and the mathematical operation would be: 3 x 3 x 3, so the same number is repeated 3 times. Just as square numbers, we have perfect cube numbers.

Try to solve the following exercises:

- What is the cube number of 7, respectively 8? What is the sum of the two cube numbers of 7 and 8?
- Identify the numbers of the following cube numbers: 512, 1728 and 3375.

You can always check **online** for lists of cube numbers.

The **perfect cube formula** is: N = X3

### Did you know that …?

- The same digit of the number will also be the last digit of the cube of that particular number. Except the following pairs: 2 will change to 8 and 8 will change to 2, and 3 will change to 7 and 7 will change to 3;
- The cube number of odd numbers will always be odd, and the cube number of an even number will always be even;
- Perfect cube numbers can be both positive or negative integers;

Now that we’ve mentioned cubes, take a break from mathematics and play with your own Rubik, or if you don’t have one, try to solve an online Rubik, **here****.**

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