Professor Dave here, let’s talk about exponents.
By now we have the four basic arithmetic operations down, those being addition, subtraction, multiplication,
and division.
But of course, there will be many more to learn in this series.
The next operation we will learn is how to raise some number to a power, or exponent.
When we do this, we have a base, which is written normally, and then an exponent, which
is written as a superscript.
Just the way that multiplication represents repeated addition, exponents represent repeated
multiplication.
Two times three is really two plus two plus two, while two to the third power is two times
two times two.
When we raise something to the second power, we are squaring that number, so we can say
five squared.
When we raise something to the third power, we are cubing that number, so we could say
five cubed.
In each case, we can write down the base number, the number of times indicated by the exponent
and do the multiplication, which gives us twenty-five and one hundred twenty-five respectively.
Sometimes instead of the superscript, we might see this little hat symbol here, and when
we see this, the first number is the base and the second number is the exponent, so
this is five to the fourth power, or six hundred twenty-five.
This is typically used when someone has written the expression without access to the superscript
function, and this symbol is needed to imply exponentiation.
One thing we will notice immediately is that with exponents, numbers get very big very
fast.
To see exactly how big, let’s examine an old Indian legend, which goes like this.
A wise man invented the game of chess, which the king immediately adored.
He liked it so much that he offered the wise man any reward of his choosing.
Being a shrewd mathematician, his request went as follows.
He said, I am a modest man, I desire but a few grains of rice.
Take this chessboard and place one grain of rice on the first square, then two grains
on the second, and then four grains on the third.
Continue in this manner, always doubling the previous number, until all the squares are
full.
The king began to oblige, but he very quickly found how impossible this request was to fulfill.
By the end of the first row, he was already up to one hundred and twenty-eight grains.
By the end of the second row, over thirty-two thousand.
The third, more than eight million.
The fourth, over two billion grains of rice.
And that’s just half the board.
On the final square, the king would have to place over a quintillion grains of rice, which
is a one followed by eighteen zeroes.
That’s enough rice to cover the entire country of India with a layer of rice a meter high,
and this much rice could not exist in the entire world.
Luckily for the king, the wise man allowed him to pay in installments.
So this legend is an example of exponential growth.
To find the number of grains that must be placed on each square, we simply raise two
to a particular exponent.
We start with one, which we could think of as two to the zeroeth power, since anything
raised to an exponent of zero will be equal to one.
Later in this clip, we will see why this is the case.
Then two to the one, or simply two, two squared, which is two times two, or four, then two
cubed, or two times two times two, which is eight.
Each time we increase the exponent by one, we are doubling the previous number, because
we are multiplying by two exactly one additional time.
This continues until we get to the final square, which is two raised to the sixty-third power.
So we can quickly see the immense power of exponents.
Let’s try a few examples to practice.
What is three to the fourth power?
That would be three times three times three times three, or eighty-one.
What about six to the fifth power?
Well that will give us five sixes to multiply, which equals seven thousand seven hundred
and seventy-six.
Can we raise something to a negative exponent, or only positive?
We can do either, in fact, and when raising to a negative exponent, we simply invert the
term and make the exponent positive, so two to the negative two is one over two squared,
or one fourth.
Since two squared is four, and four and one fourth are the inverse of one another, this
demonstrates precisely what negative exponents do.
Another way to understand this is to learn the rules of manipulating terms raised to
exponents.
X to the A times X to the B is equal to X to the A plus B.
This must be true, because let’s say we have X squared times X cubed.
That could be written as two X’s times three X’s.
But that’s now just five X’s in a row, so we should be able to call this X to the
fifth.
And that’s why we do A plus B, because two plus three is five, and that represents the
total number of times we are multiplying X.
For division, it’s the opposite.
X to the A over X to the B equals X to the A minus B. Again, this makes sense if we do
something like X to the fifth divided by X to the third.
If we change this into five X’s over three X’s, we can see that three of the X’s
cancel, leaving just two on the top, or X squared.
The number of X’s left is the number of them that are on the top minus the number
of them that are on the bottom, hence X to the A minus B.
This also explains how negative exponents work.
If we have X to the fifth over X to the seventh, five X’s cancel, and we have one over X
squared.
But five minus seven is negative two, so this must be equal to X to the negative two as
well.
This rule also tells us why anything raised to the zero power must be one.
X cubed over X cubed is X to the three minus three, or zero.
But it is also the case that all the X’s cancel, leaving us with one.
So anything to the zero power must be equal to one.
Lastly, if we do X to the A power, which is then raised to the B power, we get X to the
A times B. let’s say we have X to the fourth to the second.
That means we have four X’s, and then that whole thing is squared, meaning two sets of
four X’s.
That’s eight X’s total, or X to the eighth, which is four times two.
So these expressions summarize the rules for manipulating terms that are raised to exponents.
Let’s notice that X to the A plus X to the B is not listed here, because there’s nothing
we can do with that.
Those are two separate terms that can’t be combined in any way, so don’t make this
common mistake.
I think you’ve got the hang of it, so let’s check comprehension.