# Powers

Let's calculate $2\cdot 2=4$ and also $2 \cdot 2 \cdot 2=8$.These multiplications are simple and rapid to write, but it is not always like that. Let's see what happens if we want to multiply $2$ by $2$ seven times. We will have to write $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2=128$. In this case we realize that it is longer to write the operation.

That's why we use a much more practical notation: the exponents. In this way the number that has to be multiplied by itself is written while the number of times that is multiplied is written as a superindex. This way we indicate the number of times that we want to multiply it by itself.

For instance,

If we want to multiply the number $5$ by itself $6$ times, we will write: $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5= 5^6$.

Therefore, since $2 \cdot 2=4$ we can write $2^2=4$, and we will read "two raised to the power two (or, simply, two squared) is equal to four". Or also $4 \cdot 4 \cdot 4 \cdot 4 = 4^4$ "four raised to four", or $134 \cdot 134 \cdot 134 = 134^3$, "hundred thirty four raised to three".

This way, we have for example,

$3^5= 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$ so we avoid writing the product in such a long and extensive way. In this case we will read "three raised to the power five" which means that we multiply $3$ by $3$ five times.

In an expression like $a^n=b$ where $a$, $b$ and $n$ are natural numbers, it means that $a \cdot a \cdot\overset{(n)}{\ldots}\cdot a=b$ and we distinguish different elements.

• $a$ is the base of the power.
• $n$ is the exponent of the power
• $b$ is the $n$-th power of $a$ (when $n$ is $2$ it is said to be squared, and it is $3$ is said to be cubed).

Let's see some examples:

$7 \cdot 7=7^2=49$ where $7$ is the base of the power, $2$ is the exponent and $49$ is the square of $7$.

$2^8=256$ where $2$ is the base, $8$ the exponent and $256$ is the eighth power of $2$.

Let's see now some special powers: $0^1=0, 0^2=0 \ldots$ since, no matter how many times we multiply zero by itself, it will always be zero. $1^2=1\cdot 1=1, 1^3=1\cdot 1\cdot 1=1 \ldots$ since, no matter how many times we multiply one by itself, it will always be one. $3^1=3, 8^1=8, \ldots$ and this applies to any number with exponent $1$ since it means doing nothing. For any number it is satisfied that: $a^1=a$

We must bear in mind also, that by convention it is established that for any number it is fulfilled that: $a^0=1$. And so, $4^0=1, 345^0=1, 78^0=1, \ldots$