Exponentiation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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: x^y. Sometimes it is not possible. Then people write powers using the ^ sign: 2^3 means 2^3.

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

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

Examples:

  • 5^3=5\cdot{} 5\cdot{} 5=125
  • x^2=x\cdot{} x
  • 1^x = 1 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 a^2. So

x^2 is the square of x

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

x^3 is the cube of x

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

x^{-1}=\frac{1}{x}

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

2^{-3}=\left(\frac{1}{2}\right)^3=\frac{1}{8}

If the exponent is equal to \frac{1}{2} then the result of exponentiation is the square root of the base. So x^{\frac{1}{2}}=\sqrt{x}. Example:

4^{\frac{1}{2}}=\sqrt{4}=2

Similarly, if the exponent is \frac{1}{n} the result is the nth root, so:

a^{\frac{1}{n}}=\sqrt[n]{a}

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

a^{\frac{p}{q}}=\sqrt[q]{a^p}

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:

x=\lim_{n\to\infty}x_n

like this:

a^x=\lim_{n\to\infty}a^{x_n}

There are some rules which help to calculate powers:

  • \left(a\cdot b\right)^n = a^n\cdot{}b^n
  • \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n},\quad b\neq 0
  • a^r \cdot{} a^s = a^{r+s}
  • \frac{a^r}{a^s} = a^{r-s},\quad a\neq 0
  • a^{-n} = \frac{1}{a^n},\quad a\neq 0
  • \left(a^r\right)^s = a^{r\cdot s}
  • a^0 = 1,\quad a\neq 0: Where the base is greater than 1 and the exponent is 0, the answer is 1. If the base and exponent are both 0, the answer is undefined.

It is possible to calculate exponentiation of matrices. The matrix must be square. For example: I^2=I \cdot I=I.

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, is is not commutative. For example 2³=8 but 3²=9.

Inverse Operations[change | change source]

Addition has one inverse operation: the subtraction. Also, multiplication has one inverse operation: the 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={\textstyle \frac{3}{2}}. This is the same if you have 2 · x=3: You also get x={\textstyle \frac{3}{2}}. 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 = {\textstyle \sqrt[2]{3}}. 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).