Exponents
Exponents can make your math problems a lot easier to handle.
Simply put, it is a shortcut for multiplying numbers over and over again.
Look at the following multiplication problem:
8 × 8 × 8 × 8 × 8 × 8
Instead of multiplying 8 six times by itself, we can just write 8
^{6} and it will mean the same thing.
When reading 8
^{6}, we say eight to the sixth power or eight to the power of six.
In a similar way,
12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 = 12
^{10}
In 12
^{10}, 12 is called the base and 10 is called the exponent.
Other examples of exponents:
5
^{3} = 5 × 5 × 5
9
^{4} = 9 × 9 × 9 × 9
7
^{2} = 7 × 7
6
^{6} = 6 × 6 × 6 × 6 × 6 × 6 × 6
2
^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
In general,
y^{n} = y × y × y × y ×...× y × y (n times)
In one of the examples above, it says 7
^{2} = 7 × 7
What would you say 7
^{1} is equal to?
For 7
^{2}, you wrote down 7 twice.
Therefore, for 7
^{1}, you will write down 7 once and of course there is no need to have a multiplication sign.
7
^{1} = 7
Common pitfalls to avoid when working with exponents:
What is 2
^{6} equals to?
Is it equal to 2 × 2 × 2 × 2 × 2 × 2 ?
Or is it equal to (2 × 2 × 2 × 2 × 2 × 2) ?
It is equal to (2 × 2 × 2 × 2 × 2 × 2) = (2
^{6}) =  64
However, (2)
^{6} is a different story.
(2)
^{6} = (2) × (2) × (2) × (2) × (2) × (2) =  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2
Notice that  ×  = +
So,  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 = + × + × + × 2
^{6}
Note that + × + = +
(2)
^{6} = +2
^{6}
In (2)
^{6}, the exponent is even. Change it to any odd number and the answer will be negative.
(2)
^{7} = (2) × (2) × (2) × (2) × (2) × (2)× (2) =  ×  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 × 2
 ×  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 × 2 = + × + × + ×  × 2
^{7}
+ × + × + ×  × 2
^{7} = + ×  × 2
^{7}
Note that + ×  = 
+ ×  × 2
^{7} = 2
^{7}
Important observations about exponents:
 a^{n} is always negative.
 (a)^{n} is either negative or positive. It is positive if n is an even number. It is negative if n is an odd number.
 a^{n} is not always equal to (a)^{n}
 a^{n} may equal to (a)^{n} only when n is an odd number.
What if the base is a fraction?
When the base is a fraction it is common to use parentheses as shown below:
The above is also equal to
8
/
27
Exponents quiz to check your understanding of this lesson.

Jan 12, 22 07:48 AM
This lesson will show you how to construct parallel lines with easy to follow steps
Read More
Enjoy this page? Please pay it forward. Here's how...
Would you prefer to share this page with others by linking to it?
 Click on the HTML link code below.
 Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.