kidzsearch.com > wiki

# 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]x[/math] and [math]y[/math], the exponentiation of [math]x[/math] and [math]y[/math] can be written as [math]x^y[/math], and read as "[math]x[/math] raised to the power of [math]y[/math]", or "[math]x[/math] to the [math]y[/math]th power".^{[1]}^{[2]} 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]2^4[/math].

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

To calculate [math]2^4[/math], one simply multiply 4 copies of 2. So [math]2^4=2 \cdot 2 \cdot 2 \cdot 2[/math], and the result is [math]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]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]

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

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

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

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

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

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

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

- [math]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 *x*th power, we use an infinite sequence of rational numbers (*x _{n}*), 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 make the calculation of exponents easier:^{[3]}

- [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. In this case, 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 that x = [math]\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=log_{2}(3).

## Related pages

## References

- ↑ "Compendium of Mathematical Symbols" (in en-US). 2020-03-01. https://mathvault.ca/hub/higher-math/math-symbols/.
- ↑ Weisstein, Eric W.. "Power" (in en). https://mathworld.wolfram.com/Power.html.
- ↑ Nykamp, Duane. "Basic rules for exponentiation". https://mathinsight.org/exponentiation_basic_rules.