Cubes and Cube Roots
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Cubes and Cube Roots
In arithmetic and algebra, the cube of a number n is its third power  the result of the number
multiplied by itself twice:
n3 = n x n x n.
This is also the volume formula for a geometric cube with sides of length n, giving rise to the name.
The inverse operation 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 onethird power.
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example
23 = 8 or (x+1)3. A perfect cube (also called a cube number, or sometimes just a cube) is a number
which is the cube of an integer.
If a and b are two natural numbers such that a3 = b, then b is called the cube of a. If the units digit of
a3 is b, then the cubes of all numbers ending with a will have their units digit as b. The cubes of all
numbers that end in 2 have 8 as the units digit. The cubes of all numbers that end in 3 have 7as the units
digit. The first odd natural number is the cube of 1. The sum of the next two odd natural numbers is the
cube of 2. The sum of the next three odd natural numbers is the cube of 3, and so on.
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The cube root of a given number is a number, which, when multiplied with itself three times, gives the
number. If a given number is a perfect cube, then its prime factors will always occur in groups of three.
The cube root of a number can be found using the prime factorisation method or estimation method.
Cubes in number theory ; There is no smallest perfect cube, since negative integers are included. For
example, (4) x (4) x (4) = 64. For any n, (n)3 = (n3). Unlike perfect squares, perfect cubes do
not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where
only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect
cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last
two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube
numbers are also square numbers, for example 64 is a square number (8 x 8) and a cube number (4 x 4
x 4); this happens if and only if the number is a perfect sixth power. It is, however, easy to show that
most numbers are not perfect cubes because all perfect cubes must have digital root 1, 8 or 9.
Moreover, the digital root of any number's cube can be determined by the remainder the number gives
when divided by 3:
If the number is divisible by 3, its cube has digital root 9;
If it has a remainder of 1 when divided by 3, its cube has digital root 1;
If it has a remainder of 2 when divided by 3, its cube has digital root 8.
Waring's problem for cubes ; Every positive integer can be written as the sum of nine (or fewer)
positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be
written as the sum of fewer than nine positive cubes:
23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.
Fermat's last theorem for cubes ; The equation x3 + y3 = z3 has no nontrivial (i.e. xyz 0)
solutions in integers. In fact, it has none in Eisenstein integers.
Both of these statements are also true for the equation[2] x3 + y3 = 3z3.
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