Hi, I’m Rob, welcome to Math Antics!
In this lesson, we’re gonna learn a bit more about roots and how you can simplify them.
Have you ever noticed that teachers can be kinda picky sometimes?
Like if you give them an answer that’s not in the ‘form’ they want,
you might lose a point even if the answer is technically correct.
That’s partly because in math,
there are many different ways to write the exact same number or expression,
but some are much more clear and helpful than others.
Take the number 1 for example.
You could write it as the fraction 2 over 2,
or you could write it as the expression 1+ 0.
You could even write it as the square root of 1 if you wanted to.
The possibilities are endless!
But if it’s the answer to your problem,
why would you use any of these more complicated forms, when you could just write 1?
Isn’t that a lot simpler?
It sure is!
And math teachers always like it when answers are written in the simplest form possible.
So let’s suppose you do a math problem and get the answer: the square root of 16.
You could just leave it like that and it wouldn’t be wrong,
but can you think of a simpler way to write it?
Of course. 16 is a perfect square. It equals 4 x 4, so the square root of 16 is just 4.
That’s definitely a simpler answer.
But what if you do a different problem and end up with the answer: the square root of 32?
Well, 32 isn’t a perfect square,
so we can’t simplify it to a nice whole number like we could with the square root of 16.
You might consider using a calculator to convert it to a decimal value,
but in this case, we’d end up with an ‘irrational number’
which is a never-ending, never-repeating decimal.
And that’s definitely not a simpler way to write it.
So what other options do we have?
Well, some of you may notice that 32 could be factored into 16 times 2, right?
And we already know that 16 is a perfect square. But, does that help us out?
Actually it does because of a particular rule about square roots.
That rule says:
If you have two square roots that are being multiplied together,
like the square root of 2 times the square root of 3,
you can combine them like this: the square root of (2 x 3).
AND you can go the other way too and ‘un-combine’ them.
So if you start with the square root of (2 x 3),
you could change it to the square root of 2 times the square root of 3.
We can use that rule to our advantage in our current problem
because we just figured out that the square root of 32 is the same as the square root of (16 x 2)
which means we could re-write it as the square root of 16 times the square root of 2.
And as we already know, the square root of 16 can be simplified to just 4.
So that gives us 4 times the square root of 2, or just “4 root 2” as the simplified version.
Pretty cool, huh?
Hold on a second!
How is that answer simpler than what you had before?
Before you just had the square root of 32, and now you have a whole number times a root.
That seems even more complicated to me!!
Yeah, I know what you mean.
Sometimes it’s hard for people to decide what the ‘simplest’ or most helpful form is,
so mathematicians rely on what are called “conventions”
where they all agree on a ‘preferred’ way to express things.
Oooo, that sounds fun! …a convention for math! …like “Math-Con”!
Hey… Hey… Hey…
Would you like a selfie with the Math Antics guy?
Oh… that’s okay… thanks anyway.
Um... I’m actually looking for the Khan Academy booth…
Oh… there it is!
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Well, it’s true that the word “convention” often refers to
a group of people meeting together in a big building,
but when I say convention, I mean it’s a ‘conventional’ or ‘standard’ way of doing something.
Hey… wanna see a picture of me and Sal Khan?
Anyway… in math, it’s conventional to simplify a root if you can.
Simplifying a root means identifying any factors under the radical sign that, if you took the root of them,
would simplify to a whole number which you could then bring out in front of the radical sign.
Even though the result might seem less simple than before,
if the part under the radical sign has been made as simple (or as small) as possible,
then you’ve simplified the root.
This definition of simplifying would apply to any type of root:
square roots, cube roots, 4th roots, and so on.
But for the next couple examples, we’ll just focus on square roots
since they’re the kind you’ll encounter most often.
In the case of square roots, to simplify you’ll need to find out
if there’s any perfect squares ‘hiding’ in the number under the radical sign.
You remember what a perfect square is, right?
It’s just what you get when you multiply a whole number by itself:
like 2*2 or 5*5 or 30*30.
And how would you find out if there are any perfect squares hidden that number?
The key is to factor it!
If you’ve forgotten how factoring works, you can watch our previous videos about it for help.
Factoring will reveal if there’s any pairs of the same factor being multiplied together…
like 2*2 or 3*3.
If there are, you’ve found a perfect square ‘hiding’ in that number and you can simplify it.
Like in this example, the prime factorization of 180 is 2*2*3*3*5,
so you can re-write it like this and look for perfect squares.
2*2 is a perfect square (since its value is 4).
And 3*3 is a perfect square (since its value is 9).
Now remember the rule I mentioned earlier that says
you can re-write a square root as a product of the square roots of its factors.
That means we can re-write the problem like this.
Do you see the advantage of doing that?
Now we can simplify the square roots of the perfect squares that we found so that they just become whole numbers.
The square root of (2*2) simplifies to 2
and the square root of (3*3) simplifies to 3.
That gives us 2*3 times the square root of 5.
The square root of 5 can’t be simplified any further.
5 is a prime number, so its only factors are 1 and itself.
Great! Now all we have to do is re-combine the factors we simplified.
2*3 = 6, so the simplified version of the square root of 180
is 6 times the square root of 5 or “6 root 5”.
Let’s try another example to make sure you understand whats’a happening…
Let’s simplify the square root of 72.
If we factor 72 all the way down to its prime factors, we get 2*2*2*3*3.
As you can see, there are two pairs of factors that form perfect squares: 2*2 and 3*3
And there’s a 2 left over that doesn’t form a pair.
Like before, we could re-write this using our rule about multiplying roots.
But now that you know how that rule works, you can eliminate some of the in-between steps
if you realize that any pair of identical factors that are under the square root sign
will simplify to become a single factor out in front of the root sign.
That means you can just change the 2*2 under the root sign into a 2 out front of it,
and the 3*3 to a 3 out in front.
That gives us 2*3 times the square root of 2 or “6 root 2” as the simplified answer.
So that’s basically all there is to simplifying square roots.
And even though square roots are the most common, sometimes you may need to simplify other roots too.
For example, what if you need to simplify the cube root of 72 instead?
Well, you’d start the process the same way.
You’d factor the number under the root sign down to its prime factors to see if any parts of it can be simplified.
But since we are working with a cubed root this time,
that means we aren’t looking for pairs (or perfect squares) anymore.
Instead we need to find factors that are in groups of three.
In other words, we’re looking for “perfect cubes”.
Just like before, 72 factors down to 2*2*2*3*3.
But since we’re dealing with a cube root this time,
the 2*2*2 can be simplified because it’s a perfect cube,
but the 3*3 can’t because its a perfect square.
Also like before, we can use our rule to break this problem up
into the cube root of (2*2*2) times the cube root of (3*3).
This first part simplifies to 2
while the second part can’t be simplified any further.
So we just re-combine it to the cube root of 9.
There, the simplified version of the cube root of 72 is 2 times the cube root of 9.
So when simplifying roots in general,
it’s important to pay attention to which kind of root you’re dealing with
so you know which factors can be simplified and which can’t.
Basically, you just look at the index number, and then try to find groups of identical factors of that size.
For square roots, it’s groups of 2, for cube roots, groups of 3, for 4th roots, it’s groups of 4 and so on…
Hopefully that all makes sense, but if you’re still having trouble getting it,
I’d highly recommend re-watching our videos about exponents and roots
because it’s super important to fully understand how they work before you can understand how to simplify them.
Oh… and there’s one more quick think that I want to mention in this video
that has to do with ‘conventions’ about roots.
Sometimes an answer to a math problem might be a fraction with a root in the denominator,
like this one: 3 over the square root of 2.
The square root of 2 can’t be simplified,
but many mathematicians like to avoid having roots in the denominator,
especially when those roots are irrational numbers.
So, if your teacher asked you to re-write this fraction without a root in the denominator,
how would you do it?
Well, think back to when you learned how to add or subtract ‘unlike’ fractions,
which means fractions that don’t have the same denominator.
In those cases, you needed to change the fractions to ‘like’ fractions
by multiplying one or both of them by a special ‘whole fraction’.
We can apply that same idea to change the denominator of this fraction
into a regular whole number instead of a root.
All we have to do is make a ‘whole fraction’ out of that radical denominator.
In this case, that would be square root of 2 over square root of 2.
Then we multiply our original fraction by that new ‘whole fraction’ who’s value is just one.
On the top, we would get 3 times the square root of 2
and on the bottom we’d get square root of 2 times square root of 2
which by definition will just equal 2.
So these two fractions are equivalent.
They represent the exact same value,
but one has a radical in the denominator while the other has a radical in the numerator.
Neither answer would be ‘wrong’,
but if there has to be a radical (or root) in a fraction,
then mathematicians prefer to have it in the numerator as a convention.
If you’d like to research this idea further, it’s often called “rationalizing a denominator”,
because it’s a way to change a fraction so that it has an irrational numerator
instead of an irrational denominator,
which makes some mathematicians feel much better.
Alright, that’s all for this video.
Hopefully it’s given you a better understanding of some of the ‘conventions’ in math concerning roots.
Knowing about them can come in handy on tests in particular.
For example, if you solve a problem and get the answer square root of 72
but the multiple choice answers available to you are all in simplified form, like “6 root 2”,
you’ll realize that you just have to simplify your answer so you know which one to pick.
And remember, math is a subject that you can’t truly learn just by watching.
You actually have to apply it, so be sure to practice simplifying roots on your own.
As always, thanks for watching Math Antics and I’ll see ya next time.
Hey… you want a selfie with the Math Antics guy?
Learn more at mathantics.com.