The post summarized the topics:
 Laws of Exponents
 Exponents of 10
 Exponents of 2
 Exponents as fractions
Laws of Exponents
Law  Example 
x^{1} = x  6^{1} = 6 
x^{0} = 1  7^{0} = 1 
x^{1} = 1/x  4^{1} = 1/4 
x^{m}x^{n} = x^{m+n}  x^{2}x^{3} = x^{2+3} = x^{5} 
x^{m}/x^{n} = x^{mn}  x^{6}/x^{2} = x^{62} = x^{4} 
(x^{m})^{n} = x^{mn}  (x^{2})^{3} = x^{2×3} = x^{6} 
(xy)^{n} = x^{n}y^{n}  (xy)^{3} = x^{3}y^{3} 
(x/y)^{n} = x^{n}/y^{n}  (x/y)^{2} = x^{2} / y^{2} 
x^{n} = 1/x^{n}  x^{3} = 1/x^{3} 
Laws Explained
The first three laws above (x^{1} = x, x^{0} = 1 and x^{1} = 1/x) are just part of the natural sequence of exponents. Have a look at this:
Example: Powers of 5  
.. etc..  
5^{2}  1 × 5 × 5  25  
5^{1}  1 × 5  5  
5^{0}  1  1  
5^{1}  1 ÷ 5  0.2  
5^{2}  1 ÷ 5 ÷ 5  0.04  
.. etc.. 
Look at that table for a while … notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).
The law that x^{m}x^{n} = x^{m+n}
With x^{m}x^{n}, how many times do we end up multiplying “x”? Answer: first “m” times, then by another “n” times, for a total of “m+n” times.
Example: x^{2}x^{3} = (xx)(xxx) = xxxxx = x^{5}
So, x^{2}x^{3} = x^{(2+3)} = x^{5}
The law that x^{m}/x^{n} = x^{mn}
Like the previous example, how many times do we end up multiplying “x”? Answer: “m” times, then reduce that by “n” times (because we are dividing), for a total of “mn” times.
Example: x^{4}/x^{2} = (xxxx) / (xx) = xx = x^{2}
So, x^{4}/x^{2} = x^{(42)} = x^{2}
(Remember that x/x = 1, so every time you see an x “above the line” and one “below the line” you can cancel them out.)
This law can also show you why x^{0}=1 :
Example: x^{2}/x^{2} = x^{22} = x^{0} =1
The law that (x^{m})^{n} = x^{mn}
First you multiply “m” times. Then you have to do that “n” times, for a total of m×n times.
Example: (x^{3})^{4} = (xxx)^{4} = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x^{12}
So (x^{3})^{4} = x^{3×4} = x^{12}
The law that (xy)^{n} = x^{n}y^{n}
To show how this one works, just think of rearranging all the “x”s and “y”s as in this example:
Example: (xy)^{3} = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x^{3}y^{3}
The law that (x/y)^{n} = x^{n}/y^{n}
Similar to the previous example, just rearrange the “x”s and “y”s
Example: (x/y)^{3} = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x^{3}/y^{3}
Exponents of 10
“Exponents of 10” is a very useful way of writing down large or small numbers.
Instead of having lots of zeros, you show how many powers of 10 will make that many zeros
Example: 5,000 = 5 × 1,000 = 5 × 103
5 thousand is 5 times a thousand. And a thousand is 103. So 5 times 103 = 5,000
Can you see that 103 is a handy way of making 3 zeros?
Scientists and Engineers (who often use very big or very small numbers) like to write numbers this way.
Example: The Mass of the Sun
The Sun has a Mass of 1.988 × 1030 kg.
It is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg
(And very easy to make a mistake counting the zeros!)
Example: A Light Year (the distance light travels in one year)
It is easier to use 9.461 × 1015 metres, rather than 9,461,000,000,000,000 metres
Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.
Example: 3 × 10^4 is the same as 3 × 104
•3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000
calculator e notation
Calculators often use “E” or “e” like this:
Example: 6E+5 is the same as 6 × 105
•6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000
Example: 3.12E4 is the same as 3.12 × 104
•3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200
The index of 10 says …
… how many places to move the decimal point to the right.
Negative Exponents of 10
Negative? What could be the opposite of multiplying? Dividing!
A negative power means how many times to divide by the number.
Example: 5 × 103 = 5 ÷ 10 ÷ 10 ÷ 10 = 0.005
Just remember for negative powers of 10:
For negative powers of 10, move the decimal point to the left.
Exponents of 2
Here is a list of the number 2 raised to the power of every number from 0 to 100. Why did I make this? Because I have WAY too much time on my hands! You’re probably wondering what good this is. Well aside from the obvious (It allows quick lookup to 2’s powers) It can also be used to find out how many colors will be displayed on a screen. Example: If you’re computer monitor is set to display 16bit color, look at 2 raised to the 16th power, and you’ll see that the value is 65,536. That means that at your current display, you can have 65,536 colors on the screen at once. Now doesn’t that make you feel all warm and mushy inside?
Exponents of 2  Value 
0  1 
1  2 
2  4 
3  8 
4  16 
5  32 
6  64 
7  128 
8  256 
9  512 
10  1,024 
11  2,048 
12  4,096 
13  8,192 
14  16,384 
15  32,768 
16  65,536 
Exponents as fractions
what if the exponent is a fraction?
An exponent of ^{1}/_{2} is actually square root
And an exponent of ^{1}/_{3} is cube root An exponent of ^{1}/_{4} is 4th root And so on! 
General Rule
It worked for ½, it worked with ¼, in fact it works generally:
x^{1/n} = The nth Root of x
What about a fractional exponent like 4^{3/2} ?
That is really saying to do a cube (3) and a square root (1/2), in any order.
Let me explain.
A fraction (like m/n) can be broken into two parts:
 a whole number part (m) , and
 a fraction (1/n) part
So, because m/n = m × (1/n) we can do this:
The order does not matter, so it also works for m/n = (1/n) × m:
And we get this:

Some examples:
Example: What is 4^{3/2} ?
4^{3/2} = 4^{3×(1/2)} = √(4^{3}) = √(4×4×4) = √(64) = 8
or
4^{3/2} = 4^{(1/2)×3} = (√4)^{3} = (2)^{3} = 8
Either way gets the same result.
Example: What is 27^{4/3} ?
27^{4/3} = 27^{4×(1/3)} = (27^{4}) = (531441) = 81
or
27^{4/3} = 27^{(1/3)×4} = (27)^{4} = (3)^{4} = 81