Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Video: Evaluating Algebraic Expressions Involving Irrational Conjugates

Tim Burnham

Given that 𝑥 = 4 − √15/(4 + √15) and 𝑦 = −96/(√3 − √15), determine the value of 𝑥 + 𝑦 expressing your answer in simplest form.

07:13

Video Transcript

Given that 𝑥 is equal to four minus root 15 over four plus root 15 and 𝑦 is equal to negative 96 over root three minus root 15, determine the value of 𝑥 plus 𝑦 expressing your answer in its simplest form.

So we’re given an expression for 𝑥 and we’re given an expression for 𝑦, and we have to work out the value of 𝑥 plus 𝑦. Well, 𝑥 plus 𝑦 is simply the expression for 𝑥 plus the expression for 𝑦. So we just have to replace 𝑥 with the expression for 𝑥 and 𝑦 with the expression for 𝑦. Well, that’s phase one; but it’s certainly not in its simplest form.

For a start, we’ve got irrational numbers in the denominator, so we’d like to remove those to make it in its simplest form. And also we’ve said we’re adding negative 96 over root three minus root 15. And if we’re adding a negative number, then we’re just taking that number away. So that means instead of adding negative something, we could just take it away here.

Okay. So the next thing is gonna be to eradicate these irrational numbers from the denominators. But just before I do that, I’m gonna put parentheses around numerators or denominators that have more than one term in them. Now if I multiply each of these terms by one, then the calculation isn’t going to change. If I multiply something by one, then it’s value stays the same. But I’m not gonna actually multiply it by one itself; I’m gonna multiply it by a different version of one.

Now for the first term, the version of one that I’m gonna multiply it by is the irrational conjugate of the denominator. And by that, that’s four plus the square root of 15. So I’m gonna multiply it by four minus the square root of 15 over four minus the square root of 15. Don’t be confused that the numerator happened to contain four minus the square root of 15. Anyway, that’s kind of irrelevant here. The-the key thing is we look at the sign of the irrational in that term on the denominator and we change that sign. And you’ll see why that is in a moment.

So I’m gonna do the same thing for the other term as well; this was root three minus root 15. So I’m gonna multiply it by root three plus root 15 over root three plus 15 [root 15].

Well, let’s multiply out the parentheses in the first term. So we’ll start with the numerator. Four times four is 16, and four times negative root 15 is negative four root 15. Then negative root 15 times positive four is negative four root 15. And negative root 15 times negative root 15, well negative times negative makes a positive and root 15 times root 15 is just 15. So that’s positive 15.

And now the denominator, I’ve got four times four which is 16. And I’ve got four times negative root 15 which is negative four root 15. Then I’ve got positive root 15 times four which is positive four root 15. And lastly, I’ve got positive root 15 times negative root 15. Well, positive times negative makes negative, and root 15 times root 15 is just 15. So that’s negative 15.

And before we go on to multiply out the second term, let’s just look at the denominator of that first term. We’ve got 16 take away four root 15 add four root 15 take away 15. And if I start up with negative four root 15 add four root 15, I end up with nothing. So these two terms are gonna cancel out. And that’s why I multiply top and bottom of that expression by the irrational conjugate of the denominator so that these irrational numbers in the denominator would cancel each other out here. Now I’m also left with 16 take away 15 which is just one.

Now the numerator, I’ve got 16 plus 15 which is 31. And then I’ve got negative four root 15 take away another four root 15 which is negative eight root 15. And of course, all that over one. We don’t really need the over one; it’s just all that. So four minus root 15 over four plus root 15 simplifies down to 31 minus eight root 15.

Now for the second term, I’m actually not going to multiply out the numerator for now because there’s a chance it might cancel down. I’m hoping that the same thing is gonna happen on the denominator; we’re going to end up with no irrational numbers in there at all. So maybe something will cancel out. So let’s multiply out the denominators. Root three times root three is just three. Root three times root 15 is root 45. And of course they were both positive, so the result is positive. Then I’ve got negative root 15 times positive root three. So that’s negative root 45. And lastly, I’ve got negative root 15 times positive root 15. Well, negative times positive makes a negative number, and root 15 times root 15 makes 15. So I’ve got negative 15.

Now if we look at the terms in this denominator, I’ve got three, I’ve got positive root 45, I’ve got negative root 45, and I’ve got negative 15. Well, if I start off with root 45 and I take away forty f- root 45, I’m gonna get nothing. So those cancel out to make zero. So I’m just left with three take away 15 which is negative 12. And of course, my numerator is still 96 times root three plus root 15. Now twelves go into 96 eight times. So I could do some canceling. 12 divided by 12 is one, and 96 divided by 12 is eight. So we’ve now got 31 minus eight root 15 take away eight root three plus root 15 over negative one. So if I divide it by negative one, that makes that whole expression negative. So I’ve got the negative of a negative expression which makes it positive. And since it was over one, I can remove the over one.

So just writing that a little bit more neatly, I’ve got 31 minus eight root 15 plus eight times root three plus root 15. Well, I’m now gonna multiply out that second set of parentheses. And eight times root three is eight root three. And eight times root 15 is eight root 15, and of course that’s positive. And now if we look at those terms, I’ve got negative eight root 15 plus eight root 15. So if I add those two together, I’m gonna get a total of zero. So they cancel each other out.

Now it doesn’t matter whether I put 31 plus eight root three or eight root three plus 31; they both mean the same thing. They’re the expression in its simplest form. So there’s our answer: 31 plus eight root three or eight root three plus 31.