# Square & Cube Calculator

### More Calculators

Variance Calculator Derivative Calculator Percentage Calculator Surface Area Calculator Circumference Calculator Right Triangle Calculator Quadratic Equation Solver Random Number Generator Linear Equation Calculator Simple Interest Calculator Compound Interest Calculator Standard Deviation Calculator Matrix Determinant Calculator Pythagoras Theorem Calculator Matrix Multiplication Calculator## How To Use 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 9^{2} 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 n^{2}, 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 n^{2}.

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 n^{2}−(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} = 4n^{2}.

2. Squares of odd numbers are odd, since (2n+1)^{2}=4(n^{2}+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 = 4^{3} = 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=1^{3}),

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

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

4. The sum of the next four is the next cube (13+15+17+19=4^{3}),

5. The sum of the next five is the next cube (21+23+25+27+29=5^{3}) 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.