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

Find the multiplicative inverse of √6 + √7 expressing your answer in its simplest form.

04:18

Video Transcript

Find the multiplicative inverse of root six plus root seven expressing your answer in its simplest form.

So first of all, we need to know a little bit about the multiplicative inverse. But what we can say is, the product of a number and its multiplicative inverse is one. So therefore, we can also say that the multiplicative inverse of a number must be its reciprocal. So therefore, the MI, this is what I’m gonna call the multiplicative inverse, of root six plus root seven must be equal to one over root six plus root seven. And this is because one over root six plus root seven is the reciprocal of root six plus root seven. Also, if we multiplied them together, we’d get root six plus root seven over root six plus root seven. Which would give us our answer of one, which is what we’d be looking for. So we’ve now found the MI of root six plus root seven. So that’s the first part of the question complete.

But have we finished? Well, no because we also need to express the answer in its simplest form. And to enable us to do that, what we need to do is rationalize the denominator because we don’t want surds on the bottom. And in order to rationalize the denominator, what we need to do is we need to multiply the fraction, so our one over root six plus root seven, by the conjugate of the denominator. And the conjugate of an expression is where we change the sign in the middle of the two terms.

For example, if we had 𝑥 squared minus three, the conjugate would be 𝑥 squared plus three. So therefore, in our problem, we’ve got root six plus root seven. So therefore, the conjugate is root six minus root seven. So therefore, we’re gonna have one over root six plus root seven multiplied by root six minus root seven over root six minus root seven. So we’re actually gonna be multiplying both the numerator and the denominator.

The reason we’re doing that is because whatever you do to one, you must do to the other. So we’re just gonna have root six minus root seven as our numerator because one multiplied by root six minus root seven is just root six minus root seven. Now to get the expression for the denominator, what we need to do is expand the parentheses. If we multiply root six plus root seven by root six minus root seven, so therefore, for our first term, we’re gonna get six. And we get that because we use one of our surd rules. And that’s if we have root 𝑎 multiplied by root 𝑎, the result is just 𝑎. And that’s because if we did root six, for instance, multiplied by root six, this will give us root 36. Or the square root of 36 is six, so that’s why we get that result.

And then we have root six multiplied by negative root seven. So it gonna give us negative root six root seven. And then we’re gonna have plus root six root seven. And that’s because we’ve got positive root seven multiplied by positive root six. And then, finally, we’ve got minus seven. And we got that because we had positive root seven multiplied by negative root seven. So again, root seven multiplied by root seven gives us seven. And we get a negative because we multiplied a positive and a negative, which gives a negative result. Then, we can cancel out root six root seven because we’ve got negative root six root seven plus root six root seven, which would just be zero. So then we’re left with six minus seven, which just gives us negative one.

And this is why this is called rationalizing the denominator because as you can see, there are now no surds on the denominator. So we’ve just got negative one. So now we have root six minus root seven over negative one. So therefore, as we’re dividing by negative one, all we need to do is change the signs of both terms in our expression on the numerator. So, we could say that the multiplicative inverse of root six plus root seven, expressed in its simplest form, is root seven minus root six.

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