### Video Transcript

Find the multiplicative inverse of
the square root of six over 30.

Remember, the multiplicative
inverse of a number 𝑎 is the number that when we multiply it by 𝑎 gives us
one. Another way of considering this is
as the reciprocal of that number. So, if we have a number 𝑎, its
reciprocal is one over 𝑎. So, here, we need to find the
number that when we multiply it by the square root of six over 30, we get one. If we let 𝑥 be the multiplicative
inverse of the square root of six over 30, then we could say that 𝑥 times the
square root of six over 30 is equal to one.

Now, equivalently, we would achieve
this by solving this equation. We said that it is also the
reciprocal of the original number. That’s one over the number. So, it’s one over the square root
of six over 30. This doesn’t look very nice,
though. So, we’re going to recall how we
divide fractions. Really, we’re wanting to divide one
by the square root of six over 30. So, we write one as one over
one. And then recall that to divide by a
fraction, we multiply by the reciprocal of that fraction. This is sometimes called keep,
change, flip. So, 𝑥 is equal to one over one
times 30 over root six.

And if we multiply the numerators
and then separately multiply the denominators of our fractions, we get 𝑥 is equal
to 30 over root six. Now, in fact, we really didn’t need
to perform this step. Given a fraction in the form 𝑎
over 𝑏, its reciprocal is simply 𝑏 over 𝑎. But of course, it’s always good to
understand where these things come from. So, we found the multiplicative
inverse to be 30 over the square root of six.

But we’re really not finished. We need to rationalize the
denominator. In other words, we want the
denominator of our fraction to be rational. At the moment, it’s an irrational
number. The square root of six cannot be
written as a fraction where the numerator and denominator are integers. So, how do we achieve this? Well, we multiply the numerator and
denominator of our fraction by the square root of six. That’s the same as multiplying by
the square root of six over the square root of six, or just by multiplying by
one.

And in doing so, all we’re doing is
creating an equivalent fraction. 30 times the square root of six is
30 root six. Then, the square root of six times
itself is, of course, simply six. Multiplying a number by itself is
squaring it and squaring is the inverse to square rooting. So, we see that our multiplicative
inverse is 30 root six over six. Finally, we spot that both 30 and
six have a common factor of six. And so, dividing through by six, we
get five root six over one, which is simply five root six. The multiplicative inverse of the
square root of six over 30 is five root six.