We're asked to evaluate or simplify
the following radical expressions.
We're told to assume that the variables
represent any real number.
Notice the three radical expressions are all square roots
and therefore, the index is two.
And because two is even, and the radicands
contain variables of verbal expressions,
we need to remember, that if the index is even,
and the radicand has variables
of verbal expressions that simplify,
we must include an absolute value
around the simplified variables of verbal expressions
with odd exponents to assure the value is non-negative.
Looking at our first expression,
this is the principal square root,
and therefore, the result, must be an expression
that is always positive.
And we know the square root, undoes the squaring
and therefore, this simplifies to one factor of q plus one.
But again, because this expression must always be positive,
it simplifies to the absolute value
of the quantity q plus one.
Referring back to our reminder down here,
because the index is even,
and this simplified to an expression,
the quantity q plus one raised to the first power,
because one is odd, we had to include the expression
in an absolute value.
We can also rewrite this as just the absolute value
of the quantity q plus one.
Now, let's look at the value of q to confirm
why we need the absolute value.
Let's consider q equals negative four.
If we substitute q goes negative four
into the original radical expression,
we would have the square root of the quantity
negative four plus one squared,
which would be the square root
of the square of negative three.
And we know the square of negative three
is equal to positive nine.
And now let's evaluate this by using
our simplified expression of the absolute value
of q plus one.
So let's substitute negative four for q
in our simplified expression, which would give us
negative four plus one,
which would give us the absolute value of negative three
and the absolute value of negative three
is positive three.
So notice how without the absolute value though,
this should be the square root of positive nine
is equal to negative three, which is not true.
It's equal to positive three.
So here's an example of why we need the absolute value.
Next we have the opposite of the square root
of the quantity y plus three squared,
so because we have a negative in front of the square root,
the simplified expression must always have a negative value.
And because the square root undoes the squaring,
it's not just the opposite of the quantity y plus three,
it's the opposite of the absolute value
of the quantity y plus three.
Again, notice how the absolute value
is always going to be positive,
but because we have a negative
in front of the absolute value,
the result is always going to be negative,
which is what we need.
Again, because the index is even,
and the simplified expression, y plus three
is raised to the first power, which is odd,
again, we need the absolute value.
Of course, we can also write this
as the opposite of the absolute value
of the quantity y plus three.
Now for our last example,
before we try to simplify the square root,
we need to factor the radicand,
or the trinomial under the square root.
So we have the quantity y squared
minus four y plus four,
this will factor into two binomial factors.
We have y squared here, so we have a factor of y
and a factor of y in the first positions.
The second positions will be the factors of positive
four that add to negative four,
which would be negative two and negative two,
so we have the quantity y minus two
times the quantity y minus two,
which we can write as the quantity y minus two squared,
because we have two equal factors.
So we can write this as the square root
of the quantity y minus two squared.
Again, the square root undoes the squaring.
This simplifies to one factor of y minus two
but again, because the index is even,
and the simplified variable expression
is raised to an odd power, the power of one,
it simplifies two of the absolute value
of the quantity y minus two in this form
or the absolute value of the quantity y minus two.
I hope you found this helpful.