We are asked to multiply
the given radical expressions.
We determine these products,
just like we multiply two binomials.
Remember when multiplying two binomials,
we always have four products.
Using the idea of repeated distribution,
to determine this first product,
we begin by distributing the 2 square root X,
which gives us two products.
Then we distribute the positive three,
which gives us two more products,
giving us a total of four products.
This is equal to, two square root X,
times five square root X,
plus two square root X, times,
because of the minus four,
we have times negative four, and then we have plus,
and now we distribute the three,
and we have three times five square root X.
And then finally we have plus three times negative four.
Now we determine each product,
using the rules for multiplying radicals.
To determine this first product,
we multiply the two and the five outside the square root,
then we multiply the X and the X under the square root.
Two times five is equal to 10,
so we have 10 times the square root of X times X.
Then we have plus, to determine this product,
we multiply two and negative four
which equals negative eight.
This is equal to plus negative eight, square root X,
which is equivalent to minus eight square root X.
Then we have plus, to determine this product,
we multiply the three and the five.
Three times five equals 15.
So, plus 15 square root X.
And then finally, we have plus three times negative four,
which equals plus negative 12,
which is equivalent to minus 12.
Now, we simplify these square roots,
and combine like radicals.
We'll notice how the square root of X times X
will simplify perfectly, because we have two
equal factors of X, the square root of X times X,
or the square root of X squared,
simplifies perfectly to one factor of X.
This is equal to 10 X.
However, because the index on the square root is even,
the index is two, then the exponent
on the simplified variable is odd,
because X is equal to X to the first,
we must have an absolute value around the X.
This assures us, that the principle square root,
will always be positive.
And notice how the next to radicals are like radicals,
because the radical part is identical,
meaning the radicals have the same index,
and the same radicand.
So we have negative eight square root X,
and 15 square root of X, which equals seven square root X.
So we have plus seven square root X,
and we still have minus 12.
This is the simplified product
of the given radical expression.
Let's look at number two on the next slide.
Again, we have four products.
Using repeated distribution,
we distribute the three X squared, giving us two products.
And then we have two more products,
when we distribute the cube root of X squared.
This is equal to three X squared, times
two times the cube root of X.
And then we have plus three X squared times,
because of the minus one we have times negative one,
plus, and now we distribute the cube root of X squared.
So we have the cube root of X squared
times two times the cube root of X,
and then finally plus
the cube root of X squared, times negative one.
And now determine each product.
For this first product, we multiply the factors
outside the cube root.
Three X squared times two is equal to six X squared.
This is equal to six X squared times the cube root of X,
plus, here we have three X squared times negative one,
which equals negative three X squared.
So we have plus negative three X squared,
which is equivalent to minus three X squared.
Now we have plus, here we multiply the radicands,
or the variables under the cube roots.
So we still have the two outside the cube root.
And then we have two times the cube root
of X squared times X, which equals X cubed.
And then finally we have plus,
the cube root of X squared times negative one,
equals negative, the cube root of X squared.
And therefore we have minus
the cube root of X squared.
Now we simplify the cube roots,
and then combine like terms.
Well the cube root of X does not simplify,
but the cube root of X cubed will simplify,
because obviously X cubed is a perfect cube,
this simplifies perfectly to one factor of X.
So we have six X squared,
times the cube root of X,
minus three X squared.
And then plus two times one factor of X, or plus two X.
Then we have minus the cube root of X squared.
We don't have any like terms or like radicals,
and all the square roots are simplified,
and therefore this is the product
of the radical expression.
I hope you found this helpful.