We're asked to simplify
the given expressions.
We first have the square root of 192.
Because we have a square root
or because the index is two,
we need to the find the
perfect square factors of 192,
and therefore let's determine
the prime factorization
of 192.
192 is equal to
two times 96.
Two is prime.
96 is equal to
two times 48.
Two is prime.
48 is equal to six times eight.
Six is equal to two times three.
Both are prime.
Eight is equal to four times two.
Two is prime.
And four is equal to two times two.
The prime factorization contains one, two,
three, four, five, six factors
of two and a factor of three.
Let's write this as the square root
of two times two, times two, times two,
times two, times two, times three.
And now let's circle the
perfect square factors.
Well, two times two, or two squared,
is a perfect square factor here,
here,
and here.
So all of this will simplify,
and we'll be left with a
three under the square root.
Simplifying, the square
root of two times two,
well, the square root of
two squared simplifies
to one factor of two here, here, and here,
which gives us three factors
of two outside the square root.
Then we have times the
square root of three.
And two times two times
two is equal to eight,
giving us eight square root three.
Next, we have five square
root of 32 divided by eight.
Again, the first step is to determine
the perfect square factors of 32.
Let's look at the prime factorization.
32
is equal to two times 16,
and we can actually stop here
if we recognize that
16 is a perfect square
because 16 is equal to four times four.
So for this example,
we won't find the entire
prime factorization.
Let's stop here and write
the expression as five
times the square root of 16 times two,
all divided by eight.
We'll go ahead circle 16
because 16's a perfect square factor.
16 is equal to four times
four, or four squared.
So because the square root
of 16 is equal to four,
this simplifies to five times four
times the square root of two,
all divided by eight.
Before multiplying here,
notice how we can simplify.
We have a common factor of four
between the numerator and denominator.
There is one four in four
and two fours in eight,
which gives us five square root two,
all over two.
This is the simplified expression.
And now for the last
expression, we have the quantity
six plus four square
root 48 divided by 12.
Remember, we cannot simplify
across addition or subtraction,
and therefore we cannot
simplify the six and the 12 here
because of the addition.
Once again, let's begin
simplifying the square root of 48
by determining the perfect
square factors of 48.
Well, 48
is equal to,
if we wrote 48 as three times
16, we could actually stop.
Again, because 16's a perfect square.
But let's just say we wrote
48 as six times eight,
like we did before.
Six is equal to two times three.
Both are prime.
Eight is equal to four times two.
Two is prime.
And four is equal to two times two.
Both are prime.
So we can write this as
the quantity six plus four
times the square root of
one, two, three, four factors
of two and a factor of three.
This is still all over 12.
Well, here's a perfect
square factor of 48,
and here's another.
The square root of two
times two, or two squared,
simplifies to one factor
of two here and here,
which gives us six plus four,
times two, times two, times
the square root of three,
all over 12.
Well, four times two
times two is equal to 16,
which gives us six
plus 16 square root three, all over 12.
And now from here, we have
two options to simplify.
If we look at just the numerator,
six and 16 share a common factor of two.
So if we factor out
two from the numerator,
we would have two times
the quantity three plus
eight square root three,
all divided by 12.
And this form we can simplify because
we have a product here and
there's a common factor of two.
There is one two in
two and six twos in 12.
The final simplified expression
is three plus eight square
root three, all divided by six.
Again, we cannot simplify
the six and the three here
because we cannot simplify
across addition or subtraction.
But I also want to show how
we can simplify another way.
So let's write this
down here in the corner.
We have three plus eight square
root three divided by six,
and let's go back to this step here.
Another way to simplify this
would be to break this up
into two separate fractions,
where we have six over 12
plus 16 square root three over 12.
These two expressions are equivalent,
and now we can simplify
each fraction individually.
Six and 12 share a common factor of six.
There is one six in six
and two sixes in 12.
Notice how this simplifies to 1/2.
Here, 16 and 12 share a
common factor of four.
There are four fours in
16 and three fours in 12.
The expression can also be written as
just 1/2 plus four square root three
over three.
Notice how simplifying this way,
the expression does look
different, but it is equivalent.
If we were to obtain a common denominator
and add these two fractions,
we would get the quantity
three plus eight square
root three divided by six.
So either of these
expressions are considered
the simplified form of
the given expression.
I hope you found this helpful.
