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]\displaystyle{ x^y }[/math]. Sometimes it is not possible. Then people write powers using the ^ sign: 2^3 means [math]\displaystyle{ 2^3 }[/math].

The number [math]\displaystyle{ x }[/math] is called base, and the number [math]\displaystyle{ y }[/math] is called exponent. For example, in [math]\displaystyle{ 2^3 }[/math], 2 is the base and 3 is the exponent.

To calculate [math]\displaystyle{ 2^3 }[/math] a person must multiply the number 2 by itself 3 times. So [math]\displaystyle{ 2^3=2 \cdot 2 \cdot 2 }[/math]. The result is [math]\displaystyle{ 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]\displaystyle{ 5^3=5\cdot{} 5\cdot{} 5=125 }[/math]
• [math]\displaystyle{ x^2=x\cdot{} x }[/math]
• [math]\displaystyle{ 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]\displaystyle{ a^2 }[/math]. So

[math]\displaystyle{ x^2 }[/math] is the square of [math]\displaystyle{ 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]\displaystyle{ a^3 }[/math]. So

[math]\displaystyle{ x^3 }[/math] is the cube of [math]\displaystyle{ x }[/math]

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

[math]\displaystyle{ 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]\displaystyle{ 2^{-3}=\left(\frac{1}{2}\right)^3=\frac{1}{8} }[/math]

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

[math]\displaystyle{ 4^{\frac{1}{2}}=\sqrt{4}=2 }[/math]

Similarly, if the exponent is [math]\displaystyle{ \frac{1}{n} }[/math] the result is the nth root, so:

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

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

[math]\displaystyle{ 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 (x_i), whose limit is x:

[math]\displaystyle{ x=\lim_{n\to\infty}x_n }[/math]

like this:

[math]\displaystyle{ a^x=\lim_{n\to\infty}a^{x_n} }[/math]

There are some rules which help to calculate powers:

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

It is possible to calculate exponentiation of matrices. The matrix must be square. For example: [math]\displaystyle{ 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]\displaystyle{ \frac{3}{2} }[/math]. This is the same if you have 2 · x=3: You also get x=[math]\displaystyle{ \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]\displaystyle{ \sqrt[2]{3} }[/math]. However, if you have 2^x=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

• Exponent