Video Transcript
If 𝑘 equals four times the square root of two and two 𝑘 minus the square root of 18 equals the square root of five 𝑦, what is the value of 𝑦?
If we know that two 𝑘 minus the square root of 18 equals the square root of five 𝑦 and we know that 𝑘 equals four times the square root of two, we can plug that into our equation. If we multiply two times four times the square root of two, we get eight times the square root of two. And then we can notice that the square root of 18 can be simplified. This is because 18 equals nine times two. And we can break that apart to say the square root of nine times the square root of two. And we know that the square root of nine equals three.
We now have eight times the square root of two minus three times the square root of two. If we pull out the square root of two term, we’ll have the square root of two times eight minus three, which is five. And so we have the square root of two times five, or more commonly we would write that as five times the square root of two, is equal to the square root of five 𝑦. But the 𝑦-value is underneath this radical, and we need to get it out. To do that, we can square the square root of five 𝑦.
And that means we’ll have to square the left side of the equation five times the square root of two squared. That would be five squared times the square root of two squared. Five squared equals 25 and the square root of two squared equals two, which is now equal to five 𝑦. The five 𝑦 has come out of the radical. We’ll say that 50 equals five 𝑦. 25 times two equals 50. And to find 𝑦, we divide both sides of the equation by five. 50 divided by five equals 10. 10 equals 𝑦. And we can flip that around to say 𝑦 equals 10.
In this method of solving the problem, we did a lot of simplification, especially with this square root of 18. If you didn’t do this simplification, you could still solve the problem, but it would take a few additional steps. Let’s consider how we would solve the problem if we didn’t recognize that we could simplify the square root of 18. We’ll start where we have plugged in four times the square root of two in for 𝑘. We know that we could multiply two times four times the square root of two. So then we have eight times the square root of two minus the square root of 18 equals the square root of five 𝑦.
If at this point you felt stuck, but you’d knew that that 𝑦 couldn’t be in the radical so you’d have to square it and that that meant you needed to square the left side of the equation as well. You would then have eight times the square root of two minus the square root of 18 times eight times the square root of two minus the square root of 18 equals five 𝑦. You would have to foil these two factors, eight times the square root of two times eight times the square root of two. Well, eight squared equals 64; eight times eight is 64. And the square root of two times the square root of two equals two. The square root of two squared equals two, which we can say 64 times two equals 128.
After that, we have eight times the square root of two times negative square root of 18 which looks like this, negative eight times the square root of two times the square root of 18. From here, you’ll have to say the square root of two times the square root of 18 could be simplified to, say, the square root of 36. Now, you have negative eight times the square root of 36. But we know that the square root of 36 is six. So we have negative eight times six, which is negative 48. We’ll now foil the inner terms, negative square root of 18 times eight times the square root of two, which is the same value we just found. Negative eight times the square root of two times the square root of 18 equals negative 48.
And finally the last, the negative square root of 18 times the negative square root of 18 equals positive 18. Bring down what we found and bring down our five 𝑦. 128 minus 48 minus 48 plus 18 equals 50; 50 equals five 𝑦. And so we’re back to the expression that we found in our simplified form which confirms that 𝑦 equals 10.
