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, $$4^0=1, 345^0=1, 78^0=1, \ldots$$