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Exponentiation (power) is an arithmetic operation on numbers. It is repeated multiplication, just as multiplication is repeated addition. People write exponentiation with upper index. This looks like this: . Sometimes it is not possible. Then people write powers using the ^ sign: 2^3 means .

The number is called base, and the number is called exponent. For example, in , 2 is the base and 3 is the exponent.

To calculate a person must multiply the number 2 by itself 3 times. So . The result is . The equation could be read out loud in this way: 2 raised to the power of 3 equals 8.


  • for every number x

If the exponent is equal to 2, then the power is called square because the area of a square is calculated using . So

is the square of

If the exponent is equal to 3, then the power is called cube because the volume of a cube is calculated using . So

is the cube of

If the exponent is equal to -1 then the person must calculate the inverse of the base. So

If the exponent is an integer and is less than 0 then the person must invert the number and calculate the power. For example:

If the exponent is equal to then the result of exponentiation is the square root of the base. So Example:

Similarly, if the exponent is the result is the nth root, so:

If the exponent is a rational number , then the result is the qth root of the base raised to the power of p, so:

The exponent may not even be rational. To raise a base a to an irrational xth power, we use an infinite sequence of rational numbers (xi), whose limit is x:

like this:

There are some rules which help to calculate powers:

It is possible to calculate exponentiation of matrices. The matrix must be square. For example: .

Commutativity[change | change source]

Both addition and multiplication are commutative. For example, 2+3 is the same as 3+2; and 2 · 3 is the same as 3 · 2. Although exponentiation is repeated multiplication, it is not commutative. For example, 2³=8 but 3²=9.

Inverse Operations[change | change source]

Addition has one inverse operation: subtraction. Also, multiplication has one inverse operation: division.

But exponentiation has two inverse operations: The root and the logarithm. This is the case because the exponentiation is not commutative. You can see this in this example:

  • If you have x+2=3, then you can use subtraction to find out that x=3−2. This is the same if you have 2+x=3: You also get x=3−2. This is because x+2 is the same as 2+x.
  • If you have x · 2=3, then you can use division to find out that x=. This is the same if you have 2 · x=3: You also get x=. This is because x · 2 is the same as 2 · x
  • If you have x²=3, then you use the (square) root to find out x: You get the result x = . However, if you have 2x=3, then you can not use the root to find out x. Rather, you have to use the (binary) logarithm to find out x: You get the result x=log2(3).

Related pages[change | change source]