Video: Finding the Product of Two Irrational Conjugate Numbers

Express (5√10 + 3√3)(5√10 − 3√3) in its simplest form.

03:12

Video Transcript

Express five root 10 plus three root three multiplied by five root 10 minus three root three in its simplest form.

The first thing we’re gonna do is actually expand our parentheses. We’re gonna do this in the same way that we would expand algebraic expressions by multiplying the terms.

It doesn’t matter that we’re actually dealing with surds here. It’s done in the same way. So first of all, we’re actually gonna multiply five root 10 by five root 10.

And to enable us to do that, we’re gonna use this relationship, which is that 𝑎 root 𝑥 multiplied by 𝑏 root 𝑦 gives us 𝑎𝑏 root 𝑥𝑦, which gives us 25 root 100 because five multiplied by is 25 and 10 multiplied by 10 is 100.

So next we actually have negative 15 root 30. And that’s because we have five multiplied by negative three, which is negative 15, and then we also have root 10 multiplied by root three, which gives us root 30. And then we’ve got three root three multiplied by five root 10 which is gonna give us 15 root 30.

And then we multiply the last two terms in our parentheses. So we have positive root three root three multiplied by negative three root three, which is gonna give us negative nine root nine because we’ve got positive three multiplied by negative three.

Positive multiplied by negative gives us a negative. And then root three multiplied by root three gives us root nine.

Okay, great! So we’ve actually got to this stage. Now let’s start simplify. So our first term becomes 25 multiplied by 10 because we’ve got 25 multiplied by root 100. Well root of 100 is actually 10.

It’s actually worth noting at this point, we could have actually gone straight to this stage because root 𝑥 multiplied by root 𝑥 is always 𝑥. So as we had root 10 multiplied by root 10, that would just be 10 so be 25 multiplied by 10. However, I just wanted to want to show you each stage of how we actually get to and the line above.

Right! And it’s gonna be the same for the final term as well. So next we have negative 50 root 30 plus 50 root 30. So therefore, this is just gonna be equal to zero. And finally, we’re gonna have minus nine multiplied by three. And that’s because we’ve got nine multiplied by root nine.

Well root nine is just three. And we said previously, it could have been root three multiplied by root three is just three because we know that root 𝑥 multiplied by root 𝑥 is just 𝑥. So great! One more stage, we get 250 minus 27.

So therefore, we can say that five root 10 plus three root three multiplied by five root 10 minus three root three in its simplest form is equal to 223.

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