What are exponents and roots? Well, exponents are tricky little buggers when it comes to math. They are used in a ton of places from math to science and even the social sciences. The tricky part is that they seem so simple at first and then, “WHAM!” — you suddenly have all these rules and properties that you are supposed to know, especially when it comes to roots. Luckily, we are going to discuss not only what **exponents** and **roots** are but also how to make them work for you!

## Exponents

At its most basic, an exponent is a short cut for writing out multiplication of the same number. For example:

5^{3} is the same as saying 5 x 5 x 5. So, 5^{3} = 5 x 5 x 5 = 125.

Sometimes, the exponent is called a *power*. In the case of our example, 5^{3} can also be called 5 to third power. Now, there are some special ones that have their own names. One example is X^{2}. Since it is raised to the second power, you say that the value is *squared*. When it is raised to the third power, then you say that the value is *cubed*. The other ones don’t really have special names; sorry X^{4}.

### Negatives?

Just as you can have positive exponents, you can have negative ones. This is a little tricker because they mean something a little different.

When an exponent is negative, you are really looking at the reciprocal of the exponent. Basically, you are looking at one over the value of the exponent. This means that you are looking at a fraction. That is:

So in the case of 5^{-3}, you are really looking at the reciprocal.

## Exponent Rules

I said earlier that exponents are tricky little buggers sometimes. They make up for their trickiness with several handy little rules and properties. These properties allow you to manipulate exponents and roots — to not only do your homework but also get a fuller understanding of the math expression.

### Product Rules

Since exponents are just extended multiplication, it is only natural that you sometimes combine exponents together. When one expression with an exponent is multiplied by another expression with an exponent, the exponents simply add together. For example:

a^{m} ∙ a^{n} = a^{m+n}

which looks like this:

5^{2} ∙ 5^{4} = 5^{2+4} = 5^{6} = 15625

Another way to multiply exponents is when you have two (or more) different bases. In that case, you keep the exponent the same, but multiply the bases together. For example:

a^{m} ∙ b^{m} = (a ∙ b)^{m}

which looks like this:

5^{2} ∙ x^{2} = 5x^{2}

### Quotient Rules

Of course, what goes up must come down and what is multiplied can be divided. Remember that dividing exponents is the same as having an exponent in the denominator (a negative exponent). This means that the rules for dividing expressions with exponents are similar, but have a twist.

In the case of similar bases, the exponents are subtracted rather than added. For example:

which looks like this:

When the bases are different but the exponents are the same, the bases are divided but the exponent remains the same. For example:

which looks like this:

See? It’s not so bad. Now onto something a little more interesting.

### Power Rules

Sometimes it is necessary to raise an entire expression to some sort of power. For example:

In this case, the whole expression (exponent included) is being raised to a specific power. The exponent on the outside of the parenthesis means the same thing to the mathematical expression as if the inside of the parenthesis were a single value. To address this, you multiply the exponent inside the parenthesis by the exponent outside the parenthesis. For example:

which looks like this:

As an aside, when there is a coefficient and variable raised to a power, you apply the power to the coefficient when you simplify the expression. What I mean is:

Notice that the coefficient was squared. This is something that you need to keep in mind about raising whole expressions to a power.

## Roots

Now we get to a trickiest part of exponents. So far, we have only dealt with exponents that were whole numbers (i.e., integers). However, in certain mathematics and sciences, the exponents work out to be fractions. For example, it is possible to have an expression like:

It seems illogical to multiply a value or expression by a fraction, but you just need to take a look at what the fraction really means. In the case of one half, we are trying to find the value that, when raised to a power of 2, equals *a*.

This is called a *root*. Typically we write them like this:

The denominator represent the degree of the root that we are looking for. In this case, you want to determine the value, that when squared equals *a*. With values it would like like this:

It is really that simple. In the case of a root of 2, or raised to the one-half power, you call it a *square root*. If the power is one-third, then you are looking at a *cube root*.

Sometimes the fraction has a value in the numerator that is greater than one. For example:

In this case, the degree of the root is still equal to the denominator, but now you are also raising the base to the power of the numerator. Kind of like this:

With values it looks like this:

So roots are really just fractional exponents. When you think of them that way, the other rules still apply and work well.

## The Takeaways

Exponents represent the extended multiplication of a variable or expression. There are several rules that we can use to combine and manipulate exponents. When they come in fractions, they are called roots and have a special set up. Overall, exponents and roots are easily manipulated in the maths and sciences.

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