Multiplying and dividing by powers of 10
The effect of multiplying and dividing by some power of 10
We want to start getting comfortable with powers of ???10???, since we’ll be using them all the time for scientific notation. When we talk about powers of ???10???, we mean the result of raising ???10??? to some power:
???10^0=1???
???10^1=10???
???10^2=100???
???10^3=1,000???
???10^4=10,000???
etc.
Notice how the power (exponent) on the ???10??? is the same as the number of ???0???’s in the power of ???10??? (in the number to the left of the parentheses). For example, the exponent in ???10^4??? is ???4???, and there are four ???0???’s in ???10,000???.
The key thing to remember then is that, when we multiply a number by a power of ???10???, all we do is count the ???0???’s in the power of ???10???, and then move the decimal point that many places to the right in the other number, and that gives us the product.
How applying a power of 10 moves the decimal point
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Multiplying by a power of 10
Example
Find the product.
???67\times1,000???
Since we’re multiplying by a power of ???10???, we need to count the ???0???’s in the power of ???10???. There are three ???0???’s in ???1,000???, so we need to move the decimal point in ???67??? three places to the right. Since ???67??? has no decimal point (and so it looks as if there’s no way to move a decimal point to the right), we have to first put a decimal point to the right of the ???7??? (which gives ???67???.), and then (so that we’ll be able to move the decimal point three places to the right) we put three ???0???’s to the right of the decimal point.
At that stage in the process, we have ???67.000???, so when we move the decimal point three places to the right, we get
???67,000???
We can divide by powers of ???10??? just as easily. When we multiply by a power of ???10???, we move the decimal point to the right, but when we divide by a power of ???10??? (which is equivalent to multiplying by the number we would get if we raised ???10??? to the corresponding negative power), we move the decimal point to the left.
Notice the pattern in powers of ???10???:
???10^0=1???
???10^{-1}=0.1???
???10^{-2}=0.01???
???10^{-3}=0.001???
etc.
Example
Do the division.
???4.3\div100???
There are two zeros in ???100???, and since we’re dividing, we need to move the decimal point two places to the left. Since ???4.3??? has only one digit to the left of the decimal point (and so it looks as if there’s no way to move the decimal point two places to the left), we have to first put a ???0??? to the left of the ???4??? (to give us a total of two digits to the left of the decimal point).
At that stage, we have ???04.3???, so when we move the decimal point two places to the left, we get ???.043???. In any decimal number, however, we always want to have at least one digit to the left of the decimal point. Since there is no such digit in ???.043???, the part that’s to the left of the decimal point is understood to be ???0???, so we put a ???0??? to the left of the decimal point and get
???0.043???
Let’s do another example.
Example
Simplify the expression.
???510.75\times10^{-2}???
When the power of ???10??? is negative, we move the decimal point to the left by the number of places indicated by the exponent. Since the exponent is negative ???2???, we move the decimal point to the left two places, and we get
???510.75\times10^{-2}???
???5.1075???
So, regardless of the number we start with, if we multiply that number by a power of ???10???, then we move the decimal point to the right (by the number of places equal to the number of ???0???’s in that power of ???10???). If we divide a number by a power of ???10???, then we move the decimal point to the left (by the number of places equal to the number of ???0???’s in that power of ???10???).