# What is an Exponent?

### Quick Answer:

An exponent is a short-handed method of expressing repeated multiplication. Rather than writing \(5*5\) we can simply write \(5^2\). They mean the same thing -- the superscript 2 means to multiply five twice. Similarly, \(y^4\) means multiply y four times, i.e. \(y*y*y*y\).

### More Detail:

It doesn't seem all that hard to just write \(5*5\) instead of \(5^2\), but there are cases where the exponent could be quite large -- imagine writing out \(5^{25}\)! Farther down the road you'll also see that exponents can be negative, and don't even have to be whole numbers! There is far more to the exponent than simply saving time writing out multiplication. You'll also learn that many, many functions in math and in nature have a squared dependence -- that means one quantity depends on another value squared, or raised to the second power. For example, the area of a square is a function of the side length squared. Believe it or not, exponents will make things easier... just have patience!

### So, how can I work with exponents?

If you understand that an exponent represents the number of times you multiply something, you can immediately understand what happens when we multiply two variables with exponents:

### Example

Simplify this expression: \(x^2*x^6\)

Since \(x^2\) really just means \(x*x\), and \(x^6\) just means 6 more x's, we end up with 8 x's multiplied together, right? Well what is an exponent -- the number of times we multiply something! Therefore, \(x^2*x^6\) equals \(x^8\).

What did you learn from that example? When the same variable is multiplied, any exponents are *added together*. Adding to the exponent is the same as multiplying more times:

### Rule 1: \( x^a*x^b=x^{a+b}\)

There are other rules with exponents as well. If multiplying two variables adds their exponents, then division must subtract exponents! Check out this example:

Simplify: \( \large \frac{x^6}{x^3} \)

Well, remember that this is just a quick way of writing the following:

\( \large \frac{x*x*x*x*x*x}{x*x*x} \)

Hopefully you remember enough basic algebra that you know to cancel a factor that's in the numerator and the denominator. In fact, we can scratch off three of the x's, leaving just the numerator:

\( x*x*x=x^3 \)

So that gives us another rule:

### Rule 2: \( \frac{x^a}{x^b}=x^{a-b} \)

Let's introduce a few more rules of exponents quickly:

### Rule 3: \( x^1=x\)

That rule makes sense, because having just one x can't equal anything else but x, right? The next one might make a little less sense, but here it is:

### Rule 4: \(x^0=1\)

That's right - anything raised to the 0 power is 1. Here's why this is the case. With exponents, we are working with multiplication. The identity in multiplication is 1. Imagine taking rule 1, and adding in a couple 0's:

$$ x^a*x^b = x^{a+b+0+0} $$Logically, we shouldn't have changed anything by adding those zeros. Since you can rewrite it as:

$$ x^a*x^b*x^0*x^0 $$In order for that to be the same as \(x^a*x^b\), the \(x^0\) factors must equal 1.