Welcome to Introduction
to Multiplying Radical Expressions.
To multiply two radical expressions with the same index
multiply the coefficients together
and multiply the radicands together then simplify.
So looking at our notes, notice how the index n is the same
and therefore we multiply the coefficients together,
which would be p and q, the numbers outside the radicals.
Then we multiply the radicands together
which in this case are u and v,
the numbers underneath the radicals
and then we would simplify.
Looking at a couple examples here
notice how we have the square root of two
times the square root of three.
We have the product of two square roots
so the index is two so the indexes are the same.
The coefficients would be one
and therefore we can just multiply the radicands together
which in this case would be two times three.
We can tell this won't simplify
and therefore our product is square root six.
Next we have three cube root five
times four cube root seven.
Once again notice how the index is the same
and therefore we multiply the coefficients together
which in this case would be three times four.
Then we multiply the radicands together
which would be five times seven.
Once again, this is not going to simplify
and therefore this gives us 12 times the cube root of 35.
Now let's take a look at some more examples.
Here we have the square root of 15
times the square root of six.
We have the product of two square roots
where the index is two.
So the index is the same.
The coefficients are one, so one again
we can just multiply the radicands together
which would be the square root of 15 times six.
Now 15 times six is equal to 90
so we could write this as the square root of 90
and then simplify this
by determining the prime factorization of 90
but a lot of times it'll be easier
to find the prime factorization of 15 and six separately
and then just write all the prime factors
under the square root.
What I mean by that is 15 is equal to three times five
and six is equal to two times three
and therefore the prime factorization of 15 times six
notice would have one factor of two, two factors of three,
and one factor of five.
So we'd have two times three times three times five.
Remember this will simplify
if we have any perfect square factors
and because we're looking for perfect square factors
we want to circle groups of two equal factors
which we have here.
Three times three or three squared
is a perfect square factor.
So the square root of three squared or nine
is equal to three so this simplifies to three
times the square root of two times five, which is 10.
Next we have the cube root of 18
times the cube root of 60.
Again, the index is the same,
the coefficients are one, so we multiply the radicands
which would be the cube root of 18 times 60.
Again, we could find this product
and then find the prime factorization
but I think it's a little bit less work
if we find the prime factorization of 18
and the prime factorization of 60 separately
and then just combine the prime factorizations.
So 18 would be equal to two times nine,
nine is equal to three times three,
60 is equal to two times 30, 30 is equal to two times 15,
and 15 is equal to three times five.
And therefore the cube root of this product
is equal to the cube root of the prime factorization
of this product which should be the cube root
of, we do want to write these in order
from least to greatest.
So notice how we have one, two, three factors of two
which notice will be a perfect cube factor
and we also have one, two, three factors of three
and a factor of five.
Again, because we have a cube root
we're looking for perfect cube factors,
so I'll group the factors into groups of three,
which we see here and here
so this is two cubed and this is three cubed.
So the cube root of two cubed would be one factor of two.
The cube root of three cubed would be one factor of three
and then we still have times the cube root of five.
So this simplifies to six times the cube root of five.
Now we have three times the square root of 12
times five times the square root of 63.
They're both square roots
so we begin by multiplying the coefficients together
which would be three times five
and then we have the square root of 12 times 63.
Well 12 times five is equal to 15.
We do want to simplify this square root
so now we'll find the prime factorization of 12 and 63.
12 is equal to two times six,
six is equal to two times three,
63 is equal to seven times nine,
and nine is equal to three times three.
So again we do want to put these prime factors in order
from least to greatest, so we have one, two factors of two
and we have one, two, three factors of three
and a factor of seven.
So because we have a square root and the index is two
we'll put the equal factors in groups of two.
Here are two twos, that's a perfect square factor,
and here are two threes, that's a perfect square factor.
So simplifying we have the 15 here
and then we have the square root of two squared
simplifies to one factor of two.
The square root of three squared simplifies
to one factor of three and then we still have
the square root of three times seven, which is 21,
so we have 15 times two, that's 30, times three, that's 90
times the square root of 21.
Next we have -2 times the fourth root of 40 times seven
times the fourth root of 18.
So the index is the same.
We multiply the coefficients together first,
2 times 7 is -14 and then we'd have the fourth root
of 40 times 18.
And again we'll find the prime factorization
of 40 and 18 separately rather than find this product
and then determining the prime factorization.
So 40 is equal to four times 10,
four is equal to two times two,
and 10 is equal to two times five then we have 18
which is equal to two times nine,
nine is equal to three times three.
So we have -14 times the fourth root of this product
in prime factored form
where we'd have one, two, three, four factors of two
which notice will be a perfect fourth root
and then we have two factors of three
and one factor of five.
So we're looking for groups of four.
Here's a group of four twos, so this will simplify
but nothing else simplifies.
So we have -14 times the fourth root of two to the fourth
would be one factor of two
and then we have times the fourth root
of what's not circled.
Three times three times five is 45.
So notice how whatever we end up circling simplifies,
whatever we don't circle stays underneath the radical.
So we finally have -28 times the fourth root of 45.
Let's take a look at one more example.
Here we have negative square root 6
times -3 square root six.
So if it's helpful we could think of this first square root
as -1 times square root six.
In this form we multiply the coefficients together.
Well -1 times -3 is +3 and then we have the square root
of the product of the radicands
which would be six times six.
This isn't in prime factored form
but because we're looking for perfect square factors
or groups of two equal factors this will simplify perfectly.
So here we have three times the square root of six squared
would just be one factor of six.
Notice how this simplifies perfectly
because nothing else is left underneath the square root.
And therefore the final result is just +18.
Next we'll take a look at multiplying radicals
that contain variables.
I hope you found this helpful.