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The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n 6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourth power, by cubing a square, or by squaring a cube. The sequence of sixth ...
Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 2 and can be written as 3 × 3.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010).The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares. Some people refer to n 4 as n tesseracted, hypercubed, zenzizenzic, biquadrate or supercubed instead of “to the power of 4”. The sequence of fourth powers of integers, known as biquadrates or tesseractic numbers, is:
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a ...
Visually, 1729 can be found in other figurate numbers. It is the tenth centered cube number (a number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points), the nineteenth dodecagonal number (a figurate number in which the arrangement of points resembles the shape of a ...
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...