Express root negative three squared plus root 405 minus 55 over root five in its simplest form.
So, what I’m gonna do to enable me to express our expression in its simplest form is deal with each of the terms individually. So, I’m gonna begin with the first term which is root negative three squared.
Well, first of all, I’m gonna do negative three squared, and that’s gonna give us nine. So, we’re gonna have root nine, and that’s because negative multiplied by a negative is equal to a positive. So therefore, we know this is gonna give us an answer of three because root nine is equal to three. So, that’s our first term simplified. Let’s move on to our second term.
Well, for our second term, we have root 405. Well, root 405 can be split down to root 81 multiplied by five. That’s because we got a rule for surds, or roots, that tells us that root 𝑎𝑏 is equal to root 𝑎 multiplied by root 𝑏. And when we’re trying to simplify a surd, the first number within the root, we always want it to be a square number, and actually, the largest square number, if possible.
So, for instance, we know that 81 is a square number. And it’s the largest square number that goes into 405. So, we’ve broken root 405 down to root 81 root five. And the reason we did this is because we know if it’s a square number, then it’s gonna give us an answer when we have root of the square number. So, root 81 is equal to nine. So, we get nine root five. So, this means we’ve now simplified the second term.
Now to simplify the third term, what we want to do is rationalize the denominator. So therefore, this means making it so that we don’t have a root as the denominator. And to do this, what we do is multiply by the root itself, so root five. And whatever we do to the denominator we’ve got to do to the numerator. So, we’ve got 55 over root five multiplied by root five over root five. Then we’re gonna have 55 root five over five. And we have five as our denominator.
Because of a relationship we know with surds, and that is that if you have root 𝑎 multiplied by root 𝑎, it’s equal to root 𝑎 squared, which is, therefore, equal to 𝑎. So, in our case, if we think about that, we had root five multiplied by root five. Well, root five multiplied by root five would be root 25. And the square root of 25 is just five. So therefore, if we divide 55 root five by five, we’re gonna get 11 root five. Which means we’ve simplified the third term.
Great, so, now what we can do is pull it all together to find a simplified form for our expression. So, now we have three, because that was the simplified form of root negative three squared, plus nine root five, which was the simplified form of root 405, minus 11 root five, cause that was the simplified form of 55 over root five.
And this will give us three, then minus two root five. And that’s because if you have positive nine root five and take away 11 root five, we get a negative two root five. So therefore, we can say that if we express root negative three all squared plus root 405 minus 55 over root five in its simplest form, the answer is three minus two root five.