This video is going to be the first in a series
on simplifying radicals.
In this first part we'll look at the product rule
and the quotient rule of radicals
and some examples of how to apply them.
And in the next couple of videos we'll look
at more specific techniques for simplifying square roots
of numbers and variables.
And then [inaudible] of numbers and variables.
I want to start by reminding you of a couple rules of exponents
that we're familiar with.
A power of a product is equivalent to taking the power
of each factor separately and then multiplying.
And a power of a quotient is equivalent to taking the power
of the numerator and the denominator separately before
It's important to remember that this only works
for multiplication and division.
It does not work for addition and subtraction.
You cannot distribute this exponent to each term.
And the way I like to remember that is to think of a plus b
to the second power where you have to multiply
that binomial times itself.
And when you distribute or foil, you get this middle term 2ab.
And that's missing if you try to just distribute the exponent
to the two terms separately.
So, these are not equal.
And the higher the power,
the more middle terms you'll be missing.
Now, if you remember the relationship between the root
and the exponent, we defined the root to be the opposite
or inverse operation of the exponent.
So, it's not surprising we get a similar pair
of rules for radicals.
So, if you're taking the root of a product, you can take the root
of each factor separately.
If you're taking the root of a quotient, you can take the root
of the top and the bottom separately.
The only requirement is that the index stays the same.
So, these would have to all be square roots or all cube roots.
And just like before, we cannot distribute the radical
to the separate terms when we have addition or subtraction.
This would be incorrect.
And I'll show you an example of why that doesn't work.
So, let's start with a couple examples of quotients.
And then we'll move to some examples of products.
[ Silence ]
We'll just rewrite the rule here using a square root this time.
[ Silence ]
I want to mention here that this rule can be used
in either direction.
So, you might start with something, a square root
of a quotient and want to split it up top and bottom.
That's sort of the way I've been saying it the whole time.
But you may just as well start with a quotient
of two square roots and want to combine it
under a single radical.
So, maybe have the square root of 12 divided
by the square root of 3.
Well, these aren't perfect squares.
But in this case applying this quotient rule
and combining the 12 over 3 under a single radical,
this becomes the square root of 4.
12 divided by 3.
And that conveniently is a perfect square.
So, we're able to simplify that to a 2.
So, that shows us of one example of starting with a quotient
of radicals and combining them into the radical of a quotient.
Let me show you the other direction.
Suppose you had the square root of a quotient.
So, I want to put something in top and bottom here.
And maybe you have a perfect square in one of the top
or the bottom or maybe both.
And splitting it up then would allow you
to at least simplify part or all of it.
Let me use 25, 36.
If you're good at fractions, you can figure out the square root
of this without having to split it up.
But it's certainly a little bit easier.
Just look at the square root of the top
and the bottom separately.
This becomes 5 over 6.
So, that shows each direction.
They won't always work out this nicely.
But that makes for nice examples.
Let's look at a product now.
So, the square root of a product is equal to the product
of the first time the product of the second.
An example of this, I'm going
to pick two perfect squares, 9 and 4.
If I looked at the square roots of those separately,
the square root of 9, the square root of 4.
Those are easy enough to evaluate.
We'd get 3 times 2, which is 6.
So, there'd be no real reason to apply any rule to change this.
Since that was easy to evaluate.
But let's check it, check the rule
in this particular case and see if it works.
I guess I'm going, again, we can go in either direction.
So, it looks like I'm going to the left again here
because I'm starting with the product of 2 square roots.
I want to combine that to the square root of 9 times 4.
9 times 4 is 36.
And that is also equal to 6 just like product on the left.
So, we definitely did not prove this rule for all cases.
But we showed it true for this case.
Maybe you don't have two perfect squares.
What if you have square root of 12 times square root of 3?
Those are not perfect squares.
In this case, multiplying 12 times 3 first gives us a
So, we're able to evaluate that.
So, these examples, just like the quotients I set
up were somewhat contrived to work out nicely
with these perfect squares.
I am going to show you an example in a second
where it's not a perfect square
and the best we can do is simplify it a little bit.
And that will be, you know,
I think mainly what we use this for.
Before I do that, however, an important note here.
If I have a sun of square roots,
I want to just take this opportunity to illustrate
that this would not be the same as the square root of the sum,
in this case, 9 plus 4.
This one is easy to check.
3 plus 2 is equal to 5.
The square root of 9 plus the square root of 4.
3 plus 2 is 5.
Square root of 9 plus 4 is the square root of 13.
That's not a perfect square.
So, I can't give you an exact value for that.
But it's between 3 and 4, right?
Because 13 is in between 3 squared and 4 squared.
9 and 16. So, it's 3 point something.
All I care is that it's not equal to 5.
So, these are definitely not equal.
Here's another example.
The square root of 50.
Where's the product you might ask?
Well, I don't have one yet.
But, and in the next video I'm going to show you in detail how
to choose the factors here.
But you'll at least agree with me I hope that this is equal
to 25 times 2, the square root of 25 times 2.
And if we apply our product rule.
[ Silence ]
That is equal to the square root
of 25 times the square root of 2.
Or 5 times the square root of 2.
So, we would say that 5 times the square root
of 2 is the simplified form of the square root of 50.
You might ask why is that any better.
And that's a good question.
I'll go try to answer that in one second.
Let me just illustrate the same concept with a cube root.
32 is not a perfect cube.
But I can write it as a product
of a perfect cube times something else.
And any time you can factor the number as a perfect cube
or a perfect square, it will allow you to simplify this
by using this product rule.
[ Silence ]
So, this would be simplified.
So, why is that helpful?
One reason we like to write radicals
in their simplified form is
because of something we call like radicals.
Here's some examples.
5 square root of 2.
3 square root of 2.
These are like radicals because the radical part is exactly
The square root of 2.
The square root of 2.
Another example, 11 times the square root of 7.
Negative 5, square root of 7.
Square root of 7.
Square root of 7.
You can have cube roots.
Cube root of 5.
4 times the cube root of 5.
So, these are all examples of like radicals.
Just for comparison, if you had the square root of 2
and the square root of 3,
because the expression inside the square root is different,
these are not like radicals.
Another example, if we had the square root of 5
and the cube root of 5 because the index is different in each
of these, these are not like radicals.
So, why do we care about like radicals?
Why do we make this distinction?
Well, when we have an expression with like radicals,
we can usually simplify it further.
If you remember how we combined like terms with variables,
5x plus 3x, we added the 5 and the 3
and we got 8 times x. You can think of that as a short cut
for factoring the x out and just adding the 5 and the 3.
But it works exactly the same way when you add 5 square root
of 2 plus 3 square root of 2.
We're going to get 5 plus 3 or 8 square root of 2.
So, we can simply further if we end up with a pair
of like radicals in our expression.
Let me show you an example where we start
with some unlike radicals.
We would not be able to combine these.
But notice what happens
if we factor perfect squares out of these numbers.
9 out of the 18, 16 out of 32, 4 out of 12.
When we use the product rule for the radicals
to take the square root of each of these factors separately,
we get the simplified form of each of those radicals.
And now notice we do have a pair of like radicals here
that can be combined 7 root 2 plus 2 root 3.
And at this point, we're done because we have no more
like radicals to provide.
And everything has been simplified.
And we'll look in the next couple of videos
in a little bit more detail about this simplifying process
for square roots and cube roots.