# Exponents

## Exponent Basics

Using exponents allows you to simplify multiplying something by itself. Here is a simple example of an exponent:

${\displaystyle 6^{3}=(6)*(6)*(6)}$

An exponent is referred to as raising something to a power, in the above example, it can be referred to as 6 to the power of 3.

#### Multiplication

When multiplying exponents together using variables as shown below, you can simply add exponents to simplify.

${\displaystyle (x^{4})(x^{3})=x^{7}}$

Using an exponent on another exponent as shown below however, results in a simple bit of multiplication:

${\displaystyle (x^{3})^{2}=(x^{3})(x^{3})=(x)(x)(x)(x)(x)(x)=x^{6}}$

Here is an example of exponent addition:

${\displaystyle 3^{3+4}=3^{7}}$

#### Subtraction

Here is an example of exponent subtraction:

${\displaystyle 3^{4-3}={\frac {3^{4}}{3^{3}}}}$

#### Power of 0

Another important note, when using exponents, anything to the power of 0 is automatically 1:

${\displaystyle x^{0}=1}$
${\displaystyle 44^{0}=1}$

#### Fractional

Applying exponents to fractions is also fairly straightforward:

${\displaystyle \left({\frac {1}{2}}\right)^{2}={\frac {1^{2}}{2^{2}}}}$

## Negative Exponents

A negative exponent causes the base number with the exponent to become the reciprocal of the base itself:

${\displaystyle x^{-1}={\frac {1}{x^{1}}}}$

Another way to look at negative exponents is as though a fraction is being multiplied by a fraction as many times as the exponent shows:

${\displaystyle 3^{-4}=(((1/3)/3)/3)/3={\frac {1}{81}}={\frac {1}{3^{4}}}}$.

## Exponential Notation

A number displayed in exponential notation is basically compressing a number to it's significant parts. Here is an example:

${\displaystyle 10,000,000=1\times 10^{7}}$

More Examples:

${\displaystyle 34,545,600,000,000=3.45456\times 10^{12}}$

The secret to switching a number from exponential notation (also referred to as scientific notation) to a standard number, is just remember where the decimal point is and how many powers the 10 is raised to.

${\displaystyle 320=3.2\times 10^{2}}$

If we take the number 1,300, and imagine there is a decimal point on the far right of it. Now, we know that 10 will go into this an even number of times, so we start by counting from that decimal place, over until we get between the 1 and 3, that's 3 hops, then we multiply this number by 10³ and we have our number converted into scientific notation.

${\displaystyle 1300=1300.0=1.3000\times 10^{3}}$