Exponentiation

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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: [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]\frac{3}{2}[/math]. This is the same if you have 2 · x=3: You also get x=[math]\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]\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).

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