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:
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 our example 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 not 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}
+ × + = +
+ 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}
+ × - = -
+ × - × 2
^{7} = -2
^{7}
Observation:
-a
^{n} is always negative
(-a)
^{n} is either negative or positive. It is positive is an is an even number. It is negative if n is an odd number
-a
^{n} is not always equal to (-a)
^{n}
-a
^{n} is 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 to use parentheses as shown below:
The above is also equal to
8
/
27
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