Exponentiation

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

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

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

Examples:

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

<math>x^2</math> is the square of <math>x</math>

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

<math>x^3</math> is the cube of <math>x</math>

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

<math>x^{-1}=\frac{1}{x}</math>

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

<math>2^{-3}=\left(\frac{1}{2}\right)^3=\frac{1}{8}</math>

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

<math>4^{\frac{1}{2}}=\sqrt{4}=2</math>

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

<math>a^{\frac{1}{n}}=\sqrt[n]{a}</math>

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

<math>a^{\frac{p}{q}}=\sqrt[q]{a^p}</math>

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:

<math>x=\lim_{n\to\infty}x_n</math>

like this:

<math>a^x=\lim_{n\to\infty}a^{x_n}</math>

There are some rules which help to calculate powers:

  • <math>\left(a\cdot b\right)^n = a^n\cdot{}b^n</math>
  • <math>\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n},\quad b\neq 0</math>
  • <math>a^r \cdot{} a^s = a^{r+s}</math>
  • <math>\frac{a^r}{a^s} = a^{r-s},\quad a\neq 0</math>
  • <math>a^{-n} = \frac{1}{a^n},\quad a\neq 0</math>
  • <math>\left(a^r\right)^s = a^{r\cdot s}</math>
  • <math>a^0 = 1</math>

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

Commutativity

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

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=<math display="inline">\frac{3}{2}</math>. This is the same if you have 2 · x=3: You also get x=<math display="inline">\frac{3}{2}</math>. 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 = <math display="inline">\sqrt[2]{3}</math>. 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