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
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