Square & Cube Calculator

Lets first understand, what are squares and cubes and how we can calculate the squares and cubes of real numbers?

Square

A square number is an integer that is the square of an integer; In other words, Square is the product of some real numbers with itself.

For example, 81 is a square number, since it equals 9² and can be written as 9×9.

The usual notation for the square of a number n is not the product n×n, but the equivalent exponentiation n², usually pronounced as "n squared".

The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1×1). Hence, a square with side length n has area n².

Square numbers are non-negative. A non-negative integer is a square number is that its square root is again an integer. For example, \(\sqrt{9}=3\), so 9 is a square number. A positive integer that has no perfect square divisors except 1 is called square-free.

Properties of square

1. The difference between any perfect square and its predecessor is given by the identity n²−(n−1)²=2n−1.

2. If the last digit of a number is 0, its square ends in 0.

3. If the last digit of a number is 1 or 9, its square ends in 1.

4. If the last digit of a number is 2 or 8, its square ends in 4.

5. If the last digit of a number is 3 or 7, its square ends in 9.

6. If the last digit of a number is 4 or 6, its square ends in 6.

7. If the last digit of a number is 5, its square ends in 5.

8. In base 10, a square number can end only with digits 0, 1, 4, 5, 6 or 9.

Odd & even square numbers

1. Squares of even numbers are even (and in fact divisible by 4), since (2n)² = 4n².

2. Squares of odd numbers are odd, since (2n+1)²=4(n²+n)+1.

3. Square roots of even square numbers are even, and square roots of odd square numbers are odd.

The sum of the n first square numbers is

$$\sum_{i=1}^n i^2=1^2+2^2+\cdots+n^2$$ $$=\dfrac{n(n+1)(2n+1)}{6}$$

Cube

The cube of a number n is its third power, it means, the result of multiplying three instances of n together.

The cube is also the number multiplied by its square:

$$n^3=n*n^2=n*n*n$$

Cube of 4 = 4³ = 4*4*4 = 64.

The cube function is the function that maps a number to its cube. It is an odd function, as \((-n)^3=-(n)^3\).

The volume of a geometric cube is the cube of its side length, giving rise to the name 'cube'.

The inverse operation that consists of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power as \(n^{\frac{1}{3}}\).

Sum of first n cubes

The sum of the first n cubes is the

$$\sum_{i=1}^n i^3=1^3+2^3+\cdots+n^3$$ $$=(1+2+\cdots+n)^2$$ $$=\left(\dfrac{n(n+1)}{2}\right)^2$$

Sum of cubes of numbers in AP

There are some examples of cubes of numbers in AP whose sum is a cube:

\(3^3+4^3+5^3=6^3\)

\(11^3+12^3+13^3+14^3=20^3\)

\(31^3+33^3+35^3+37^3+39^3+41^3=66^3\)

Cubes as sums of successive odd integers

In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13,...

1. The first one is a cube (1=1³),

2. The sum of the next two is the next cube (3+5=2³),

3. The sum of the next three is the next cube (7+9+11=3³),

4. The sum of the next four is the next cube (13+15+17+19=4³),

5. The sum of the next five is the next cube (21+23+25+27+29=5³) and so on.

For calculating squares & cubes using above square & cube calculator, you have to just write the number in the given input box and press the calculate button, you will get the result.