It’s Professor Dave, let’s learn about square roots.
We’ve learned how to solve simple algebraic equations, we just look at what is happening
to the variable we want to solve for, and we do the inverse of that operation to both
sides.
If we have X plus two, we subtract two.
If we have three X, we divide by three.
What if we have X squared equals nine?
What is the inverse of squaring something?
That would be taking the square root.
When we take the square root of X squared, we get X, because squaring X is what would
give us X squared.
On the other side, the square root of nine is equal to three, because three squared,
or three times three, gives us nine.
So taking the square root of something is like asking what number times itself will
give us the number inside the square root?
We must point out something very important here.
Negative three is also a valid answer for the square root of nine, because negative
three times negative three also equals nine.
Remember that a negative number times a negative number is positive, so both of these values
work for the square root of nine.
That means that the square root of nine is equal to positive three or negative three,
which we sometimes label this way, with both the plus and minus sign next to the three,
and we refer to this as plus or minus three.
Any square root will do this, so there are always two answers when we take a square root.
Since negative three squared equals nine, we should also note that the square root of
negative nine is not negative three.
In fact, there is no real solution to this term, so we can’t take the square root of
negative numbers the way we take the square root of positive numbers.
If we were to try to do so, we would get an imaginary answer, which is something we will
investigate later in the series.
For now, let’s just say that only positive numbers can have their square root taken.
It’s very easy to get the square root of perfect squares.
The square root of four is plus or minus two.
Continuing down the list, nine, sixteen, twenty-five, thirty-six, all give very tidy answers when
we take the square root.
We should make some effort to memorize the first ten or even twenty perfect squares,
because they will come in handy later.
But the numbers in between these perfect squares also give a square root, it just won’t be
an integer.
Some of these roots give a decimal value that goes on forever, meaning they are irrational.
We will talk more about this concept later, but for now let’s acknowledge that this
is an interesting result, because it means that there are real numbers that can’t be
expressed as fractions, no matter how big we make the denominator.
We should also note that radical expressions, as we often call them, can sometimes be simplified.
Take something like root eight.
Eight is not a perfect square, so we can’t get a neat and tidy answer when we take its
square root.
But, we can change eight into four times two.
The reason we would want to do this is that root four times two is the same as root four
times root two, and root four does work out nice and tidy, since it’s equal to plus
or minus two.
So root eight can be simplified to become plus or minus two root two.
We could say the same for root twenty-seven.
Twenty-seven is nine times three, and root nine is plus or minus three, so we get plus
or minus three root three.
Just remember to look for any factors that are perfect squares in order to get these
expressions to their simplest form.
Are there other roots we can take besides the square root?
Certainly, we can take the cube root.
This is the opposite of cubing something, or raising it to the third power.
If X cubed is twenty-seven, what is X?
We will now have to take the cube root of both sides.
This is like saying, what number raised to the third power will give the number inside
this root.
In this case, the answer is three, because three times three times three equals twenty-seven.
Also notice that in this case, this is the only answer.
Negative three doesn’t work, because negative three times negative three times negative
three equals negative twenty-seven.
So negative three is the cube root of negative twenty-seven.
In this way, cube roots differ from square roots, first because you can take the cube
root of a negative number, and second because any number will only have one unique cube
root, rather than two.
If we take the fourth root of something, we see that the rules are similar to square roots
in that we can’t take the fourth root of a negative number, because multiplying a negative
value four times gives us a positive value.
Taking the fifth root sees us going back to rules like for the cube root.
We alternate back and forth depending on whether the root is odd or even.
Odd roots have one real solution and even roots have two real solutions.
For now, we just have to accept this, but later we will get a more satisfying explanation.
We now come to one of the most important things to understand about roots, we have to realize
that they represent fractional exponents.
Taking the square root is the same as raising something to the one half power.
This is because, if the square root of something times itself should equal the original value,
like how root X times root X equals X, then it must be the case that we can express this
as X to the one half times X to the one half equals X to the one, since one half plus one
half equals one, according to the rules of exponents.
This is a notational choice, and it works for any exponent.
Taking the cube root is the same as raising something to the one third power.
We can also combine two different operations into one fraction.
Raising something to the two thirds power means squaring it and taking the cube root,
so eight to the two thirds power equals four, which we can verify by either squaring first
and then taking the cube root, or taking the cube root and then squaring.
In this way, X to the A over B power simply means to multiply X by itself A times and
then take the B root.
When we combine this with what we know about operations with terms raised to exponents,
we can do seemingly tricky things.
Take this expression here, the square root of X times the cube root of X divided by the
fourth root of X.
First let’s change this to X to the one half times X to the one third divided by X
to the one fourth.
Since X to the A times X to the B equals X to the A plus B, we simply add these exponents
together on the top.
To do so, we need a common denominator for these fractions, so we change them to three
sixths and two sixths, and add them together to get five sixths.
Then if dividing by the X to the one fourth we subtract, which means we need the lowest
common denominator, which will be twelve.
So ten twelfths minus three twelfths equals seven twelfths.
And there you have the simplified version of the original expression, X to the seven-twelfths
power.
Let’s check comprehension.