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Exponential Notation

The idea behind multiplication is to simplify repetitive addition.  For example, instead of adding 5 + 5 + 5 + 5 = 20 , we can say 5 * 4 = 20.  Of course, this "shortcut" involves learning some new rules and some new facts.  It is shorter to write and often is shorter and faster to compute this way.

When we have repetitive multiplication, we can also use a shortcut.  This is called exponential notation.  It is shorter to write and often is easier to compute.

Exponential notation involves a "base" which is the number that is being multiplied and the exponent which tells us how many times the base is used as a factor.  The base is written as a standard numeral and the exponent is written as a superscript:  bx

5 * 5 * 5 = 125.  We can write it instead as 53 and say "5 to the 3rd power."  The five in this example is the factor and in exponential notation we call it the "base" while the three is the number of times the factor is used and we call it the "exponent" or less formally, the "power."

64 = 6 * 6 * 6 * 6 = 1296                            73 = 7 * 7 * 7 = 343                        104 = 10 * 10 * 10 * 10 = 10,000

Rules for using exponents

It is easy to see that all of the numbers you normally use can be written in exponential notation.  For instance, when you write 8, it could be written as 81.

You also can easily see that you could multiply two number in exponential notation.  43 * 43 = 4 * 4 * 4 * 4 * 4 * 4 = 4096.  Once you see this you might take a second look and note that 43 * 43= 46.   This gives us a rule:  ab + ac = ab+c.  This only makes sense when you are multiplying numbers with the same base. 

You don't add the exponents if the bases are unlike.  4 * 4 * 5 * 5 = 42 * 52.

We know that when we divide a number (except 0) by itself we get one.  That helps us show another rule for exponents.  Since 51 ÷ 51 = 1 and  73 ÷ 73 = 1  and  45 ÷ 45 = 1 then we can see that instead of adding the exponents, when dividing we subtract.  That tells us that 51 ÷ 51 = 51-1 = 50   and 73 ÷  73 = 73-3 = 70 = 1 and 45÷ 45 = 45-5 = 40 = 1.

In math, when  we have a rule that works, we say that we generalize.  In general, any nonzero number raised to the zero power equals one.