Video Transcript
Determine which has a larger value:
Is it two root three or three root two?
There are two ways to answer
this. One method is to estimate the value
of the square root part first and then multiply them by two and three,
respectively. It’s much easier, though, to undo
this simplification of the surd. Let’s see what that looks like. We begin by recalling that two is
equal to the square of four. And so, if we take our first
number, two root three, we can then write that as the square to four times the
square root of three. But of course, for real numbers 𝑎
and 𝑏, the square root of 𝑎 times the square root of 𝑏 is the same as the square
root of 𝑎𝑏.
This means that two root three is
the same as the square root of four times three, which is 12. Let’s now repeat this process with
our second number. That’s three root two. This time, we recall that three is
equal to the square root of nine. And so we’re going to rewrite three
root two as the square root of nine times the square root of two. That is, of course, equal to the
square root of nine times two, which is the square root of 18.
And so let’s compare the sizes of
these two numbers. We have the square root of 12 and
the square root of 18. Now it follows that since 18 is
bigger than 12, the square root of 18 must be bigger than the square root of 12. And so the square root of 18 has a
larger value than the square root of 12, which, of course, means that three root two
must have a larger value than two root three. The answer is three root two.
