In this example, we will approximate the values
of different radical expressions by determining
which two integers the following irrational
numbers fit between.
We will do this without using a calculator.
We will state this as a three-part inequality.
In part a), we’re going to be working with
root 10, or the principal square root of 10.
The way to begin this is to first determine
which two squares does 10 fit between, specifically,
what are the two closest squares, one less
than and one greater than, 10.
Well I know 10 is less than 16, and I know
10 is greater than 9, 9 and 16 being the closest
perfect squares.
So if 10 falls between 9 and 16, then we know
the square root of 10 must fall between the
square root of 9 and the square root of 16.
By having picked perfect squares with the
9 and the 16, I can simplify this statement
to say that 3 is less than the square root
of 10, which is less than 4.
We have now approximated that 10 is somewhere
between 3 and 4.
Because 10 is closer to the 9, we can at least
estimate that the square root of 10 will be
closer to 3 than it is to 4.
We’re going to skip part b) at the moment
and skip to part c).
In part c), the expression we’re working
with is root 18 all over 2.
To begin this one, we’re going to first
focus on the 18 and find the two perfect squares
that 18 sits between.
I know 18 is greater than 16, and 18 is less
than 25, 16 and 25 being the closest perfect
squares to 18.
So because of that, I know that the square
root of 18 will fall between the square root
of 16 and the square root of 25.
Another way to phrase that same statement
is 4 has to be less than the square root of
18, and the square root of 18 has to be less
than 5.
So root 18 sits between 4 and 5.
Taking into account that we weren’t dealing
with root 18, but rather root 18 over 2, I
can go a step further.
I can divide all of those values in half.
So that would then further tell me that 2
is going to be less than the square root of
18 over 2, which will be less than 5 over
2, which is the same thing as 2 and a 1/2,
or 2.5.
So this goes a bit further than the directions
had indicated, which was to determine which
two integers these numbers fall between.
We’ve actually narrowed it down either further,
that root 18 over 2 must between 2 and 2 and
a 1/2.
If we really wanted to keep carefully to the
directions, we could take it a step further,
because we know 2 and a 1/2 is less than 3.
So we could expand it out further to say root
18 over 2 is between 2 and 3.
But honestly 2.5 is a little bit more restrictive
and a little bit more precise.
On part d), we have opposite of the square
root of 84.
Again, we’ll start by focusing on the radicand.
We want to pin 84 between two perfect squares.
I know 84 is greater than 81, but less than
100.
So that tells me then the square root of 81
is going to be less than the square root of
84, which will in turn be less than the square
root of 100.
But by picking 81 and 100, I can simplify
the statement to say that 9 is going to be
less than the square root of 84, which will
be less than 10.
So root 84 fits somewhere between 9 and 10.
That being said, we were not asked to work
with root 84, but rather the opposite of root
84.
So using what we know about inequities from
previous courses, something that truly hasn’t
come up yet in this course, is I want to actually
change the sign on all of those numbers.
I want to change this to a -9, a negative
root 84, and a -10.
One of the rules we hopefully remember is,
if you take an inequality and you multiply
both sides of the inequality by a -1, so I’ll
multiply the 9 by -1; I’ll multiply the
root 84 by -1, and I’ll multiply the 10
by -1, not only does it change the signs there,
but it also has an impact on the inequalities.
So if I need to state this as a three-part
inequality, I need to be able to say that
-9 is greater than negative root 84, which
in turn will be greater than -10.
Now having discussed that, let’s come back
to part b).
In part b), we are dealing with 1 minus root
17.
We will begin with the root 17.
By focusing on 17, we will first pin 17 between
16 and 25, similar to what we did with the
root 18 in part c).
I know 16 is less than 17, and 17 is less
than 25.
So then in turn, I can throw a root over each
one of those.
So root 16 must be less than root 17, which
in turn is less than root 25.
This will simplify down to 4 is less than
root 17, which in turn is less than 5.
So that’s focused on just the radical piece,
but now taking into account the fact that
we actually want to work with the negative
root 17, I’m going to end up multiplying
each one of those by a -1: -1 times 4, -1
times the root 17, and a -1 times the 5.
But remembering when we change the sign on
the values, we also change the sign on the
inequality, meaning the inequality gets reversed.
So another way to say that I know 4 is less
than root 17 is less than 5, I can reverse
that to say that -4 must be greater than negative
root 17, which must in turn be greater than
-5.
The original expression we had was 1 minus
root 17.
It’s important to recognize that 1 minus
root 17 is the same thing as negative root
17 plus 1.
So if I can add 1 to each one of these expressions,
-4 plus 1, negative root 17 plus 1, -5 plus
1, I’ll actually finally arrive at what
I’m trying to figure out.
-4 plus 1 is -3.
Negative root 17 plus 1 is 1 minus root 17.
And -5 plus 1 is -4.
So through several steps of work, by starting
with the radicand and pinning the radicand
between two integers, taking into account
we’re actually dealing with the negative
root, and we really wanted one more than that
negative root, I have now identified that
1 minus root 17 will be somewhere between
-3 and -4.
