In Math and Science classes there's a specific way to write really small or
large numbers. This is called scientific notation, the
way we typically write regular numbers is called standard notation.
You will be asked to convert from standard notation to scientific notation
or vice versa. So let's learn the general form for a number in scientific notation.
We always start with just one number that is not zero
this number is then followed by a decimal and ends with one or more
numbers, zeros are fine in this location, this is
all multiplied by 10 raised to a number. This number or
exponent can either be positive negative or zero.
You may also see scientific notation like this
where there is only one number multiplied by 10 raised to an exponent.
Now let's go over how you can tell if a number will have a negative
or positive exponent or an exponent of zero.
All small numbers which are numbers less than one
will have negative exponents when you are converting them to scientific
notation. For example one we are converting
0.00143 which is in standard notation to scientific notation.
Remember this is our general template for scientific notation
that we will be using. We will first move the decimal place over until we reach a
number that is not zero. So we will move the decimal to the
right one, two, three times and rewrite the
number. We now have 1.43 now since we moved the
decimal over three times this means our exponent will be negative
three. The three comes from the total number of times we needed to move the
decimal and the negative is because this is a
small number or number that is less than one.
All large numbers which are numbers greater than or equal to one
will have positive exponents when you are converting them to scientific
notation. For example 2 we'll convert this number
from standard notation to scientific notation.
Again we will use our general template for scientific notation.
We would first find the decimal place. For numbers greater than or equal to 1
the decimal place is always after the entire number on the far right we will
first move the decimal place over until we reach a number that is not zero.
So we will move the decimal to the left one,
two, three, four times and rewrite the number
we now have 6.7800 now since we moved the decimal
over four times. This means our exponent will be four. The
four comes from the total number of times we needed to move the decimal
and it's positive because this is a large number or number that is greater
than or equal to one. Numbers that already have one number
that is not zero followed by a decimal will have an
exponent of zero when converting them to scientific notation.
For example three we are converting 5.72 which is in standard notation
to scientific notation and here's our general template for scientific notation.
In this example we already have the first part of the general template
so we don't have to move the decimal place and since we didn't move the
decimal place this is why the exponent would be zero.
Now let's practice going the opposite way so converting
from scientific notation to standard notation.
It is important to know that for negative exponents we move the decimal
place to the left while for positive exponents we
move the decimal place to the right. For our first example we will convert
4.2 times 10 to the negative third power to
standard notation. Remember the exponent tells us how many
times we move the decimal over and the sign of
the exponent tells us which way to move the decimal. So since
our exponent is negative three this means we will move the decimal to
the left three times. When we do this we get 0.0042.
For our second example we will convert 8.4 times 10 to the 5th
power to standard notation. Remember the exponent tells us how many times we move
the decimal over and the sign of the exponent tells
us which way to move the decimal. So since our
exponent is positive 5 this means we will move the decimal to
the right 5 times. When we do this we get 840,000.
For our third example we will convert 6.31
times ten to the zero power to standard notation.
Since our exponent is zero this means we did not move the decimal so our answer
would be six point three one. This is the case
because any number raised to the zero power
is one. So ten raised to the zero power is
one and we essentially are multiplying six point three one
times one which is six point three one. Something to be aware of is there are
numbers that may appear to be in scientific notation but aren't
really in the proper form. You will need to know how to change
these types of numbers into the proper scientific notation
so let's practice. For this first example 26.3 times 10 to the fourth
seems like it's in scientific notation but remember the template is that we
must only have one number in front of the decimal not
two. So we will have to move the decimal over
once to the left. For these types of questions where there
already is a 10 to an exponent we will follow these specific rules. When
we move the decimal to the left we would add by however many times we
needed to move the decimal over to have just one number be in front
of the decimal. So in this case we only move the decimal over one time to
the left so we will add one to the initial exponent of four
giving us five so the proper scientific notation
is this. Now when we move the decimal to the right
we would subtract by however many times we needed to move the decimal
over to have just one number be in front of the decimal.
0.041 times 10 to the negative third power
requires us to move the decimal two times to the right
so we will subtract a 2 from the initial exponent of negative 3 to get a negative
5. The proper scientific notation is this.
This next part is good to know if you aren't allowed to use a scientific
calculator. Alright before we start going over how
to perform these operations with numbers in scientific notation
remember the typical scientific notation template looks like this.
We will refer to the number in front as the coefficient
the 10 is called the base and this number is our exponent or
power. When multiplying or dividing two numbers that are in scientific notation
it is important to know three main exponent rules.
The first rule is whenever our bases are the same and are being multiplied
we add the exponents together to get our new answer.
The second rule is whenever our bases are the same and are being divided
we subtract the exponents together to get our new answer.
The third rule is whenever a base is raised to another power or exponent
then we will multiply the exponents together to get our new answer.
Let's apply these rules to numbers in scientific notation
when multiplying two numbers in scientific notation we
first will multiply the numbers in front or coefficients together.
In this example we will multiply 2.6 times 3.7
to get 9.62. Now that we multiply the coefficients we will move on to the
bases. Our first exponent rule applies here
because we have the same bases of 10 and the bases are being multiplied so we
must add the exponents together to get our new answer.
Putting this together with our new coefficient and new base
and exponent we get 9.62 times 10 to the 9th power. When dividing two
numbers in scientific notation we first will divide the coefficients
together. So we will divide 3.6 by 1.2 to get 3.
Next we'll move on to the bases, our second exponent rule
applies here since we have the same base of 10 and the bases are being divided
so we must subtract the exponents together to get our new answer.
Putting this together we get 3 times 10 to the sixth power.
When a number in scientific notation is raised by another power or exponent
we first will raise the coefficient by that power.
So 1.4 will be raised to 3 which means we are multiplying 1.4
three separate times by itself to give us 2.744.
Now that we found our coefficient we can move on to the base.
The third exponent rule applies here since the base is raised to an
additional power of 3. So we will multiply these exponents
together to get our new exponent of 15. Putting this all together our answer is
2.744 times 10 to the 15th power. When adding
and subtracting numbers in scientific notation we first have to make sure the
exponents on the bases are the same number. If they are not we
must make them the same number by moving the
decimal to the left or to the right. Remember that when a number is in
scientific notation or even appears to be in scientific
notation the following rules apply. If we were to move the decimal to
the left we would add by the number of times
we moved the decimal over to the initial exponent.
So if we move this decimal over once to the left we would add a 1
to the initial exponent to get our new answer.
If we move the decimal to the right we would subtract by the number of times we
move the decimal over from our initial exponent. So if we
move this decimal over twice to the right
we would subtract by 2 from the initial exponent to get this as our new answer.
Let's do an example. So right away we should see that the exponents are not
the same number so we have to change one of these
exponents by moving the decimal on our coefficient.
I typically change the first number in scientific notation to match the
exponent of the second number in scientific notation
so if we want to change this exponent of 3 to 2
we would have to subtract by 1. Remember that if we move the decimal to the right
we would subtract so we'll move the decimal over one time
to the right and subtract by 1 from our initial exponent and get this
as our new value. Now that both numbers have the same
exponent we can add the coefficients together.
52.1 plus 6.14 is 58.24 we do not add the bases or exponents
here we simply bring 10 to the second down. Finally this
number is not in the proper scientific notation
because we always want just one number that is not zero in front of the decimal
so we will move the decimal over to the left.
Since we moved the decimal one time to the left we must add a one to the
exponent to give us our final answer of 5.824 times 10 to the third power.
Let's do another example. Before we subtract these numbers we see that the
exponents are not the same number so we have to change one of these
exponents by moving the decimal in our coefficient.
Let's change this negative seven into a negative five by moving the decimal over
two times to the left. When we move the decimal over twice we
are adding by two to our original exponent
so negative seven plus two is negative 5.
Now that both numbers have the same exponent we can subtract the
coefficients together. 0.0337 minus 4.97 gives us a negative
4.9363, we do not subtract the bases or
exponents we simply bring 10 to the negative fifth down and this
is our answer. Since you don't want to waste your time
and retake chemistry I recommend doing more practice problems.
That's exactly why I've created them for you and these all come with step-by-step
video answers. Simply click the link in the description
to try them out