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: 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

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, it 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).