Exponentiation

In mathematics, exponentiation (power) is an arithmetic operation on numbers. It can be thought of as repeated multiplication, just as multiplication can be thought of as repeated addition.

In general, given two numbers [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], the exponentiation of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] can be written as [math]\displaystyle{ x^y }[/math], and read as "[math]\displaystyle{ x }[/math] raised to the power of [math]\displaystyle{ y }[/math]", or "[math]\displaystyle{ x }[/math] to the [math]\displaystyle{ y }[/math]th power". Other methods of mathematical notation have been used in the past. When the upper index cannot be written, people can write powers using the ^ or ** signs, so that 2^4 or 2**4 means [math]\displaystyle{ 2^4 }[/math].

Here, the number [math]\displaystyle{ x }[/math] is called base, and the number [math]\displaystyle{ y }[/math] is called exponent. For example, in [math]\displaystyle{ 2^4 }[/math], 2 is the base and 4 is the exponent.

To calculate [math]\displaystyle{ 2^4 }[/math], one simply multiply 4 copies of 2. So [math]\displaystyle{ 2^4=2 \cdot 2 \cdot 2 \cdot 2 }[/math], and the result is [math]\displaystyle{ 2 \cdot 2 \cdot 2 \cdot 2=16 }[/math]. The equation could be read out loud as "2 raised to the power of 4 equals 16."

More examples of exponentiation are:

• [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]

Similarly, 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 power is simply the reciprocal of the base. So

[math]\displaystyle{ x^{-1}=\frac{1}{x} }[/math]

If the exponent is an integer less than 0, then the power is the reciprocal raised to the opposite exponent. For example:

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

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

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

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

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

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

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

In some cases, 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_n), 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 make the calculation of exponents easier:

• [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. In this case, 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 that 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 that x=log₂(3).

Exponentiation Media

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  Graphs of y = b'x for various bases b:*  base 10,*  base e,*  base 2,*  base 1/2.*Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.

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  Power functions for n=1,3,5

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  Power functions for n=2,4,6

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  From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8.

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  The limit of e^1/n is e⁰ = 1 when n tends to the infinity.

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  The three third roots of 1

Related pages

• Exponent
• Exponential function
• Exponentiation by squaring
• Tetration