Question Video: Finding the Multiplicative Inverse of Real Numbers Involving Roots

Find the multiplicative inverse of (√6)/30.

02:54

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.

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