In this lesson we want to investigate multiplying by a power of ten. The goals
are to evaluate ten raised to a power, and then also to multiply by a power of ten.
Let's first see if we can identify a pattern for powers of ten.
Well then to the first would be equal to ten.
Ten to the second would be ten times ten, which equals one hundred.
Ten to the third, we'd multiply three factors of ten, which is equal to a thousand.
Ten to the fourth would be four factors of ten, which is equal to ten thousand.
And ten to the fifth would be five factors of ten, which is equal to 100,000.
If we count the number of zeros here, notice it matches the exponent on ten.
So in general, ten to the power of n in expanded form is one followed by n zeros.
So if we had ten to the eighth, this would be one followed by eight zeros,
which would be one hundred million.
Now let's take a look at multiplying by a power of ten.
Five times ten to the second is the same as five times one hundred,
which is equal to five hundred. Two times ten to the third
is the same as two times one thousand, which equals two thousand.
Nine times ten to the fourth would be nine times ten thousand, or ninety thousand.
And before we discuss the pattern, let's look at two more.
Fourteen times ten to the second would be fourteen times one hundred,
or fourteen hundred, or one thousand four hundred.
And two hundred seventy-three times ten to the third would be two hundred seventy-three times one thousand.
Notice one shortcut would have been just add the number of zeros that match the exponent.
Here we just added two zeros to five,
here we added three zeros to two,
here we added four zeros to nine and so on.
But we also want to develop a pattern that we could use when multiplying a decimal
times the power of ten. For example, if we take a closer look at this
product here,
the decimal point on this nine would be just to the right, or right here.
So if we had nine, when we move the decimal point to the right, the same
number of places as the power on ten, so in this case four places to the right,
we could also determine this product.
So we move this to the right four places. One, two, three, four,
and then fill in the missing digits with zeros,
notice we also have the product of ninety thousand.
So when multiplying by a power of ten with a whole number exponent,
we can just move the decimal point to the right, the same number of places as
the exponent.
Notice for fourteen times ten to the second, if we have fourteen,
the decimal point there,
move the decimal point two places to the right,
we would have one thousand, four hundred.
So let's try a couple of these when multiplying a decimal times a positive power of ten.
Here we have zero, point, five times ten to the second,
which would be the same as zero, point, five times one hundred.
So we could do this the long way, but the shortcut is to take the decimal
zero, point, five,
and then move the decimal point two places to the right.
One, two, and then fill in the missing digits with zeros,
and this would give us our product of fifty.
So for one, point, two, seven times ten to the third,
which would be equivalent to one, point, two, seven times one thousand.
We'll take our decimal of one, point, two, seven,
and move the decimal point three places to the right.
So starting here we'll move it one, two, three places.
We're missing one digit here,
so this product will be one thousand, two hundred seventy.
Let's take a look at two more of these.
We have twenty-seven, point, three, four times ten to the fifth.
Which would be the same as twenty-seven, point, three, four times
one, followed by five zeros.
But to determine this product we'll take the decimal of twenty-seven, point, four,
and move the decimal point five places to the right. So one,
two, three, four, five.
We're missing three digits, so we'll add three zeros. One, two, three.
Putting in our commas here, we would have two million, seven hundred, thirty-four thousand.
And let's take a look at one last example.
One, point, two, seven, five times ten to the fourth is
equivalent to one, point, two, seven, five times one, followed by four zeros,
or ten thousand. So we'll take our
decimal of one, point, two, seven, five and move the decimal point four places to the right.
So one, two, three, four.
We're missing one digit, so the product would be twelve thousand, seven hundred fifty.
So this is a nice shortcut when multiplying by a positive power of ten,
and the exponent is a whole number.
I hope you found this helpful.