Exponents for simplifying products
Exponents help us write numbers in a simpler way
Exponents are a tool we can use to write numbers in a simpler way. An exponent is a little number that you write above and to the right of another number, like this:
???3^2???
When you see an expression like this, the little ???2??? is the exponent, and the ???3??? is called the “base.” The exponent tells you the number of times to multiply the base by itself. So the expression ???3^2??? is telling you to multiply ???3??? by itself ???2??? times, since the base is ???3??? and the exponent is ???2???. Here are some others:
Let’s expand these examples just to be clear what we really mean. When we say to multiply ???3??? by itself ???2??? times, we mean that
???3^2=3\cdot3=9???
In the same way, multiplying ???2??? by itself ???3??? times means that ???2^3=2\cdot2\cdot2=8???. And multiplying ???5??? by itself ???4??? times means that ???5^4=5\cdot5\cdot5\cdot5=625???.
Exponents are really helpful to us as we go further in math, because if we want to express a multiplication like ???7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7??? in a simpler way, we can use an exponent and write it as ???7^{11}???.
What exponents are and what they do
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Rewriting an expression using an exponent
Example
Use an exponent to write the expression.
???4\cdot4\cdot4\cdot4\cdot4\cdot4???
In this expression, we’re multiplying ???4??? by itself ???6??? times. Which means we need the base to be ???4??? and the exponent to be ???6???. Therefore, we can write the expression as
???4^6???
Let’s do another example, but this time we’ll take an exponential expression and expand it.
Rewriting an expression with an exponent in expanded form
Example
Write the expression in expanded form.
???6^3???
This expression tells us to use ???6??? as a factor ???3??? times, which means we can rewrite it without an exponent as
???6\cdot6\cdot6???
We don’t have to find the result of the multiplication, but we could also do that and say
???6^3=6\cdot6\cdot6=216???
We can use exponents with variables as well. So if we want to multiply ???x??? by itself ???3??? times, we can write that as ???x^3???:
???x\cdot x\cdot x=x^3???