This is the third exponent discussion. So, from the first discussion we know that:
24 * 22 = 26
and more general am * an = a(m+n)
AND, also, we know:
94 / 92 = 92
and more general am / an = a(m-n)
Also, we know
(94)2 = 98
and more general (am)n = a(m*n)
And also, even more basically we know
94 = 9*9*9*9 and more general an = a*a*a*a ... n times
One final thing we know is that any number raised to the zeroth power is 1.
That is to say:
90 = 1
and more general a0 = 1
Never, under-estimate the value of being able to multiply and divide by 10, 100, 1000, etc FAST.
Almost no thinking, do it FAST. If you have a number you need to multiply by 10, just
move the decimal place of that number one time to the right. We move it
one time because there is one zero in 10. We move it to the right because we are multiplying.
Similarly with no thinking, if you have a number you need to divide by 10, just
move the decimal place of that number one time to the left. We move it
one time because there is one zero in 10. We move it to the left because we are dividing.
OK, following along 17 * 10 = 170
143 * 10 = 1430
Lets multiply by 1000 (there are 3 zeroes)
17 * 1000 = 17000
143 * 1000 = 143000
But what about 14.5 times 1000, same rule
14.5 * 1000 = 14500 remember, if you can't see the decimal it is after
the last number . 14500 is the same as 14500.
And if you need commas
14,500 is the same as 14,500.
15 * 10 = 150 the decimal in 15 is at the very right 15. And 15. is the
same thing as 15.0 so move it one time to the right giving 150. Several more.
9 * 10 = 90 same as 9.0 * 10 = 90
127 * 10 = 1270 same as 127.0 * 10 = 127
7.5 * 10 = 75
38.5 * 10 = 385
OK, got it. One time to the right when multiplying by 10. By the way, 10 has one
zero in it. Guess what if you multiply by 100? Two times to the right. Such as
9 * 100 = 900 same as 9.00 * 10 = 900
7.5 * 100 = 750 38.5 * 100 = 3850
1.1 * 100 = 110 0.5 * 100 = 50 just two times to the right
Well, how bout multiplying by 1000? Yep, three places to the right. If you
are multiplying you move it right. How many places you move it is the number
of zeros you have. Multiply by 1,000,000? Move it 6 to the right.
7.5 * 1000000 = 7500000
38.5 * 1000000 = 38500000
How about dividing by 10? Count zeros = 1. Move decimal to the left by one. 7.5 / 10 = 0.75 and
38.5 / 10 = 3.85
Divide by 1000? Move it 3 places (there are 3 zeros there) to the left.
7.5 / 1000 = 0.0075
38.5 / 1000 = 0.0385
Now, this multiplying and dividing by 10, 100, 1000, etc has great value
in scientific notation.
To quote wikipedia,
"Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in standard form"
Remember that word ... "conveniently".
Scientific notation was not developed to confuse our lives but to make handling numbers be convenient.
You might have trouble believing that at first, but it is the truth. In scientific notation we write numbers
as a*10b and most of the time we do not use the asterick but rather a multiplication "x sign" as in
the wikipedia article.
I can hear you saying under your breath, "Where is the convenience here ... I'm not 'feeling the love'".
Consider this number
349,000,000 that is a big number. In scientific notation, that number would be
3.49*108 and you read that as "3 point four nine times 10 to the 8th".
In scientific notation you always have the decimal point after the first non-zero digit (3 in this case).
So, how do you figure this out. Well, it has to do with a couple of things. First, multiplying by one.
Second, all that "10, 100, 1000 stuff" above.
So, take 349,000,000 and I wish to write it as "3.49 * 10 to the some power". What power do I use?
Notice that decimal point between the 3 and the 4. Originally it was here 349,000,000. < --
It has been moved 8 times to the left. How do you move it 8 times to the left. Well, divide it by 100000000.
Check it out. BUT, you can't just willy-nilly go around moving decimal points. If you are going to divide by
100000000 then you also have to multiply by 100000000, or else you would have a different number.
For goodness sakes if you divide by 100000000 and multiply by 100000000 then you are really multiplying by one
and the number is the same. Anything multiplied by 1 is itself.
So, I am going to use my "dividing by 100000000" to get my decimal point here 3.49 and then I have to multiply by 100000000,
which is the same as 108.
So, the answer of "what is 349,000,000 in scientific form" is 3.49*108
Its kinda "as easy as that" but then again that was a lot of work. Lets see if we can find an easier way to think about this.
Use this thought process and see if it is faster.
Take 349,000,000 we need the decimal at 3.49 to do that we moved the decimal 8 times to the left.
We have to "counteract" (nullify/neutralize) this movement by multiplying by 108 .... therefore
349,000,000 => 3.49*108
You still may not be "feeling the love" relative to scientific notation. Its use has, in great part,
something to do with "significance". If you had $3 in your pocket and I wanted to borrow $1, you might not let me.
That $1, after all is very significant. However, if you had $300,001 in your pocket (or a briefcase),
you would have a much greater tendency to let me borrow it. After all, compared to three hundred thousand,
one is not very much. Well, you see that wikipedia states that this has great value to "engineers", among other people.
I can speak to "engineers" to some degree, since I am one. If we worked an engineering problem and we got 349,008,321
as a preliminary answer which we needed to use later, we might be very likely to round it off to 349,000,000.
Looking at that 8, it is in the thousands position. Eight thousand compared to 349 million just is not very "significant".
So, we would have a clear tendency to drop the 8321 part of that number and treat it as 349 million, which in scientific
notation is 3.49*108.
Scientific notation is a practical, pragmatic tool. Say you were building a new house and were told it would cost you
$110 per square foot and you knew the house you wanted was 2200 square feet. So, REAL FAST, what would the costs of
the house be? You need a calculator? NO! Consider this:
2200 = 2*1100= 2*1.1*103
110 = 1.1*102
So, our answer is
2*1.1*102*1.1*103
That is just 5 numbers multiplied together. Remember, in multiplying order is unimportant.
Said another way, 4*3 is the same thing as 3*4. So, I choose to multiply them in this order
2*1.1*1.1*102*103
and I think I will start at the right-hand side. We know from our exponent lessons that 102*103 = 105
also we know (or should know) what eleven times eleven is. So 1.1 times 1.1 is 121 with the decimal
point then placed in the proper location => 1.21 . So now we have 2*1.21*105.
Multiplying by 2 is rather easy.
The answer is 2.42*105 or (move the decimal 5 times right) 242000.
While we are at this point, there is one more thing we should mention.
Us engineers are not real wild about STRICK scientific notation as we prefer "engineering notation"
which is a close first cousin to scientific notation. Remember in scientific notation the decimal is
ALWAYS after the first non-zero number. So, in scientific notation our answer was --> 2.42*105 .
Engineers prefer the exponent of the 10 to be a multiple of 3, such as 103 which is one thousand OR
106 which is one million OR
109 which is one billion
So, rather than strict scientific notation 2.42*105,
we prefer 242*103 . Notice the decimal point moved two places to the right to
accommodate the exponent changing to a 3. And we would easily read 242*103 as 242 thousand,
rather than the strict scientific notation of "two point four two times 10 to the fifth".
This is NOT to make you negative to scientific notation, but rather is another way of looking
at its first cousin, engineering notation.
Well, assuming you still are "not feeling the love of scientific notation" because you know for a
fact you are not going to be a scientist, mathematician, or engineer and will never need to estimate
anything in your life (which is a big stretch) ... what you need to do is chalk this up to
something you have to learn for the math class you are in now (or else you would not be reading this).
So, GET OVER IT ... and feel the love of scientific notation. :)