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