# Exponent rules

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##### Intros
###### Lessons
1. Powers VS. Exponents
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##### Examples
###### Lessons
1. Write each expression as a single power. Then, calculate.
1. ${4^3}\; \times \;{4^4}$
2. ${3^2}\; \times \;{3^5}$
3. ${\left( { - 7} \right)^3}\; \times \;{\left( { - 7} \right)^3}$
2. Simplify each expression into a single power. Then, calculate.
1. ${\left( { - 3} \right)^6} \div {\left( { - 3} \right)^5}$
2. ${6^4} \div {6^1}$
3. ${\left( { - 5} \right)^6} \div {\left( { - 5} \right)^4}$
3. Rewrite each expression and then calculate.
1. Rewrite ${\left[ {2\; \times \left( { - 7} \right)} \right]^3}$ as the product of two powers. Then, calculate.
2. Rewrite ${\left( {{8^3}} \right)^6}$ as a single power. Then, calculate.
3. Rewrite ${\left( {\frac{7}{8}} \right)^6}$ as the quotient of two powers. Then, calculate.
4. Solve the following:
1. Use a pattern to evaluate $- {5^0}$.
2. Calculate$\left( { - {5^0}} \right)\; \times \;\left( { - {5^0}} \right)\; \times \;\left( { - {5^0}} \right)$.
5. Simplify each expression into a single power.
1. ${\left( {{2^2}} \right)^6}\; \times \;{2^3}$
2. $\frac{{{{\left( { - 6} \right)}^4}{{\left( { - 6} \right)}^2}}}{{{{\left( { - 6} \right)}^3}}}$
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###### Topic Notes
Numbers with power can look complicated and difficult to calculate. Luckily, we can use exponent rules to simplify those expressions for a clear look and easier calculation.

## Exponents

In math, when we deal with a number we don't know yet, we make that number a variable. The symbol for a variable can be letters such x or y.

You should already know exponents, but let's revise the exponent definition. An exponent tells you how many times a variable is multiplied by itself. So when you see $x^4$, you'll know that it's actually $x \cdot x \cdot x \cdot x$. It's a shorthand way to show multiplying a number by itself.

In the $x^4$ example, the exponent is the 4. The base is $x$. The exponent tells us how many times to multiply a base to itself. The base is, of course, the thing that's being multiplied. When we use exponents, we call it "raising to a power". The power equals to the exponent, so in our example, $x$ is raised to a power of 4.

Two special powers you should be aware of is when a number is raised to a power of 2, and when a number is raised to a power of 3. For a power of 2, that number is "squared". For a power of 3, that number is "cubed". So $x^3$ would be read as "$x$ cubed".

So when we have a variable with an exponent, are we able to simplify the expressions? The answer is yes! There's a few rules you'll have to follow so that you can properly work with exponents and they're called exponent rules. They are as follows:

Multiplying exponents with the same base

When you carry out multiplication of exponents with the same base, you add their exponents together.

For example: $x^3 \times x^4 = x^7$

Dividing exponents with the same base

When you divide exponents that have the same base, you subtract their exponents.

For example: $(3)^4 \div (3)^3 = (3)^1$

These are the two basic rules that we'll learn for now when it comes with dealing with exponents. Again, keep in mind that these rules only work on expression that have the same base. If they do not have the same base, you won't be able to simplify their exponents based on the exponent properties shown above.

## How to multiply exponents

Putting what we just learned into use, what happens when we multiply exponents with the same base of 4?

Question 1:

$4^3 \times 4^4$

$=4^{(3+4)}$

$=4^7$

We simply add their exponents together to get the final simplified answer of $4^7$

## How to divide exponents

Now let's try it with dividing exponents. In the below case, we've got a (-3) base.

Question 2:

$(-3)^6 \div (-3)^5$

$=(-3)^{(6-5)}$

$=(-3)^{(1)}$

$=-3$

When we divide exponents, we'll subtract them from one another. In this case, since we get an exponent of 1 in the end, it means we won't have to do anything to the base. Therefore, our answer is just -3.

To see the long form of writing out numbers with exponents, as well as learning how to input exponents into your calculators, check out this article. You'll eventually have to learn even more laws of exponents that will help you simplify and calculate any type of expression that has exponents in it!