Lesson video

Hi, welcome to today's lesson on applying the structure of multiplication and division with surds.

By the end of today's lesson, you'll be able to multiply and divide with surds.

Our lesson has three parts and we're going to start with part one, which is generalising the rules for multiplying and dividing with surds.

Is this statement always true, sometimes true, or never true? What do you think? Pause the video and write down your thoughts now.

Welcome back.

What did you go for? I went for it's sometimes true and the reason I've done that is that I can think of a case where the statement is true and cases where the statement is not true.

When would this be true? Well, it would be true if x has the value of zero.

If it does, then what I've written is root five equals root five.

That's definitely true.

What about this statement? In other words, when it's not true? And that would be when x is not equal to zero because then I'd have something, for example, if x was one, root six equals root five and that's not true.

Remember, surds have a unique value so it's not possible for root six to equal to root five.

Now at this point, some of you might be saying, "But hang on a second, what about if one is the positive square root and one is the negative square root?" And that's something that does sometimes come up here because you've looked at the quadratic formula and you've seen that in the quadratic formula, there are two possible solutions.

In other words, you consider the positive square root and the negative square root.

However, convention is that when we see the radical sign written as you can see now on the screen, we are referring to the positive square root.

We refer to the positive square root as the principal square root.

It's now time for your first task.

I'd like you to sort the statements you can see into three categories, the statements that are always true, the ones that are sometimes true, and the ones that are never true.

If needed, you should give some examples to justify your answers.

Pause the video and do this task now.

Welcome back.

Let's start by looking at the group that is always true.

The sum of a rational and irrational number is irrational.

Well, that makes sense, doesn't it? A rational number can be expressed as a fraction and an irrational number cannot.

Summing of rational and irrational number is not going to give us a rational answer.

It makes sense, therefore.

This is true that rational and irrational must produce an irrational answer.

Let's have a look at the next one.

Root five multiplied by root x is equal to the square root of 5x.

Well, we know that when we multiply surds, we multiply the radicands, so that makes sense, and then root x divided by root six is equal to the square root of x divided by six.

Well, we know that when we're dividing, we divide our radicands and that's all we've done there, so I'm happy that these statements are always true.

What about the sometimes true category? Well, we've got root x multiplied by root y equals the square to z.

Well, now that might be true.

For example, we could use numbers here to say this.

If x is four and y is three, then as long as z is 12, that works because four times three is 12.

What about if we chose other values though? So if we said x was nine and y was four, if z is equal to anything other than 36, that statement's false.

You might have gone further and said it's true as long as said is z equal to the product xy and that would've been fine too.

What about the next one? The product of two surds is a surd.

Well, we know that's not always true.

It's true, for example, where we had root three multiplied by root two, they gave us root six.

But if we square a surd, well then the result is the product of two surds, but because we squared a surd, we got a rational answer as you can see there with root three multiplied by root three.

What about our last one? Now you might have thought that was never true except we can come up with an example here.

It's true when x is equal to seven, seven subtract seven is zero, so what we'd have is root of zero, which is zero, is equal to the root of seven, take away the root of seven, which would be zero.

We've given some examples of when it's false here.

For example, when x is 11 and you can check that on your calculator if you like.

What's about our never true group and there's only one statement for this, that the sum of two surds is rational.

Well, that can't be true.

An irrational number plus an irrational number just gives us an irrational number.

Let's do a quick check.

I'd like you please to order the steps to show that root five multiplied by root eight is equal to two root 10.

In other words, rearrange A, B, C, and D to show the logical steps of our working.

Pause the video and do this now.

Welcome back.

Let's see how you rearranged it.

We started first of all by writing the square root signs as exponents so that instead of writing root five, I wrote five to the power of a half and I did the same with the eight.

I then said, ah, the exponents are the same and I'm multiplying so I can multiply the bases, but I can simplify that to give us 40 to the power of a half, which is the square root of 40, and the square root of 40 as we know simplifies to two root 10.

Let's now generalise our rules for multiplying and dividing.

The square root of x multiplied by the square root of y is the same as saying x to the power of a half multiplied by y to the power of a half.

And we know that we can rewrite this as x times y all to the power of a half, but that's just the same as saying xy all square rooted.

In other words, we can say root x multiplied by root y is equal to the square root of xy.

We're now going to write two division facts related to root five multiplied by root eight is equal to root 40 and we did this in the previous lesson, so use that multiplication to write two division facts involving those three surds.

Pause the video and do this now.

Welcome back.

Let's see what you put.

We should have written root 40 divided by root eight is root five and root 40 divided by root five is root eight.

Let's look at writing a general rule now for division.

When square rooting x divided by y, I can write the following.

It's equivalent to x multiplied by y with an exponent of negative one.

Now you know that's true because you saw that in your work on indices.

We can write this therefore as x multiplied by y to the negative one all to the power of a half.

Now again, we can use our laws of indices here, that's equivalent to x to the power of a half multiplied by y.

Now remember, we've multiplied those exponents so negative one multiplied by a half is a negative half.

But now we can rewrite this as x to the power of a half over y to the power of a half, or in other words, the square root of x divided by y is the same as saying the square root of x divided by the square root of y.

In the boxes on the right-hand side of the screen, you can now see our two generalised rules for multiplying and dividing with surds.

We're going to use those generalised rules now and apply them when multiplying and dividing with surds.

On the left-hand side of the screen, I've asked you to calculate root eight multiplied by root five.

Well, let's check that we understand how that would work.

Root eight multiplied by root five is the same as the square root of eight multiplied by five which is a square root of 40 which I can obviously simplify to two root 10.

Now it's your turn.

Multiply root eight by root six please.

Pause the video while you do this.

Welcome back.

Let's see what you put.

Well, root eight multiplied by root six is equal to the square root of eight multiplied by six, which is root 48, which we know can simplify to four root three.

What about with division? The square root of eight divided by the square root of two.

Well, we can write that as follows so the square root of eight divided by two or the square root of four which is two.

Now it's your turn.

Please calculate root 40 divided by root eight.

Pause the video while you have a go.

Welcome back.

Let's see how you got on.

Root 40 divided by root eight is the same as saying it's the root of 40 divided by eight.

In other words, root five, which we know can't simplify any further.

I've put the general rules for multiplying and dividing the surds on the right-hand side of the screen in case you find them useful to refer to.

What I'd like you to do please is find the answer in its simplest form.

for the square root of 3xy multiplied by the square root of six y squared.

Pause the video while you have a go at this.

Welcome back.

Let's see how you got on.

Well, when we multiply these together, three times six is 18 then we have x, and then y times y squared is y cubed, so we're left with 18xy cubed.

Now, 18 can be written as nine times two.

We know that will simplify to three root two, so three will be the coefficient and two will be part of the radicand.

X cannot be simplified because there are no square factors there so x will remain as part of the radicand.

Y cubed, however, can be written as y squared times y.

The y squared is square rooted so y becomes part of the coefficient and then the remaining y stays as part of the radicand, meaning that the square root of 18xy cubed becomes 3y multiplied by the square root of 2xy.

What's about this one? Root six divided by root 24.

Don't forget, you can use the rules on the right if you think they'll help.

Pause the video and have a go now.

Welcome back.

How did you get on? Did you spot you could simplify this.

We know that we can write root six over root 24 as the square root of six over 24.

Well, six over 24 can be simplified as a fraction.

We can divide both numerator and denominator by six leaving us with a fraction of a quarter.

So actually what we've got here is the square root of a quarter, which is simply a half.

What about this one? Square root of a quarter multiplied by the square root of three-quarters.

Pause the video and have a go now.

Welcome back.

How did you get on? Did you end up with this? When we multiply, we can multiply the radicands together.

And because it's fractions, we know we multiply the numerators, well that will give us three, and the denominators which gives us 16.

We're left, therefore, with the square root of three over 16.

Can we simplify any of that? We can simplify the denominator so the numerator will become root three and the denominator would become four.

It's now time for your second task.

I'd like you to complete the multiplication grids.

Now, although they're called multiplication grids, you might need to use some division to work out some of those missing values.

Pause the video now while you have a go.

Welcome back.

Let's see how you got on.

As you can see from our grids here, everything that is in a bold outlined box was one of the factors of the values you see in the four inner boxes.

So for example, if we look at box A, in order to work out the seven root two, I had to multiply the seven from the column by the root two in the row and that's what we've done here with all of these.

You'll notice that some of them, for example, have been simplified.

It's now time for the third and final part of our lesson and that's when we consider whether we should simplify before we multiply or divide with our surds.

It can be easier to simplify before we multiply or divide with surds.

For example, root 48 multiplied by root 27.

Now I can multiply 48 by 27.

That's actually gonna take me a bit of time.

I wonder if there's a simpler way to do it.

Let's consider what simplifying does to this problem.

Well, the square to 48 can be simplified to four root three and the square root of 27 can be simplified to three root three.

Well hang on, I can now see I've got four times three, which is 12, and root three times root three, which is three.

So actually what I've got here is 12 times three.

12 times three is just 36.

I don't know about you, but that was a lot easier than doing 48 multiplied by 27 because even if I had done that, I would then needed to have simplified the resulting surd and I don't know the square root of 36 off the top of my head.

It's now your turn.

Have a go at working out root 162 divided by root eight.

Do it by simplifying first and see how much easier that is.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

Well, the square of 162 can be written as nine root two and eight can be written as two root two.

As you can see, we can use the commutative law to rewrite this.

So we have nine multiplied by two multiplied by root two by root two leaving us with 18 multiplied by two or 36 again.

What about this one? Well, the square root of 45 can be written as three root five and the square root of six over 75 can be written as, now the square root 75 is five root three so the denominator was fairly easy.

Can you see that for the square root of six, I've actually written the square root of two multiplied by the square root of three.

Can you see why I might have done this? That's right, by doing that, I'm multiplying both numerator and denominator by root three, which means I can cancel.

I'm therefore left with three root five multiplied by root two over five giving me three lots of root 10 all divided by five.

It's now your turn.

Multiply root 32 by root 15 divided by root 50.

Again, try simplifying first to see if you can ways to make the calculation easier.

Pause the video now.

Welcome back.

Let's see how you got on.

Root 32 can be written as four root two, root 15 can be written as root five multiplied by root three and root 50 can be written as five root two.

Now, although I wrote the 15 as root five by root three, you're gonna see I didn't actually need to do that.

I did it because I thought it might be useful.

It ended up not being so if you'd left it as root 15, you did much better than me.

I then did the multiplication.

I'm multiplying by root two, but I'm also dividing by root two and since everything on the numerator was multiplied by root two and all of the denominator was multiplied by root two, I know that I can simplify.

So my numerator became four root five root three all over five, which I then tidied up into four lots of root 15 over five, but there were other ways you could have approached this to get to the same result.

It's now time for the final task.

For each of these, please perform the calculations.

Be careful because there's now some addition and subtraction in there too, but you can use all of your rules that you've covered so far in order to be able to calculate these.

Don't forget to fully simplify.

Pause the video and have a go now.

Welcome back.

Let's look at the second part of this task.

In this, we're asking you to find the missing information using your knowledge of geometry.

In other words, area with different basic shapes.

Pause the video and have a go now.

Welcome back.

Let's go through our solutions.

In the top one, suggested that you fully simplify.

So in other words, root 75 became five root three.

We were then able to say that's two root 15 add, well, three lots of five is 15 and root five times root three is root 15, which is 17 root 15.

In B, remember, we know that when we are square rooting a fraction, it's equivalent to square rooting the numerator and then dividing by the square root of the denominator.

This means that the denominator can become four and the numerator can simplify to A root five.

In C, again, I went through and simplified as much as I could.

The numerator therefore became six root two subtract four root two, which we know will simplify to two root two.

The denominator became 10 root two multiplied by two root two, which gives us 40.

We could then take out a factor of two from both numerator and denominator leaving us with root two divided by 20.

In D, I had to use my formula for the area of a trapezium.

Remember to calculate the area of trapezium, we sum the parallel sides, so root 80 plus root 45, and then multiply by the perpendicular distance between them, so multiply by root five and then we divide by two.

Do you remember why we divide by two? It's because two identical trapezium make a parallelogram and so actually what we were doing was finding the area of the parallelogram and then halving it because we only want one of the two trapezium that we used.

Starting at the top, therefore, what I've done is I've simplified both surds inside the brackets and I can see I therefore have like terms, four root five and three root five is seven root five.

I then multiplied by the root five outside and this gives us 35.

Don't forget, of course, we are dividing by two which means that our area is 17.

5.

In E, we know to find the area of a rectangle are the two perpendicular sides multiplied together.

So in other words, I know that E multiplied by root eight is equal to six root six, so I can find the length of E by doing division, six root six divided by root eight.

Well, I simplified the root eight to be two root two and then realised, of course, root six can be written as root three multiplied by root two, so that the roots of two will cancel, leaving me with six root three over two, six divided by two is three, so E must be three root three.

Our final shape is our triangle.

Now, in order to calculate the area of this triangle, I know that it's base times height divided by two.

Well, the base of my triangle is root 50, but I don't know the perpendicular height, but I can work it out using Pythagoras.

The square root of 62 all squared minus the square root to 50 all squared and then square rooted, well that gives a 62 takeaway 50, which is 12, so it's just the square root of 12.

Now that I have my perpendicular height, I can use the formula for the area of a triangle, base times perpendicular height divide by two.

Root 50 multiplied by root 12 is the same as saying five root two multiplied by two root three, or in other words, 10 root six.

And then we mustn't forget to divide by two, so my area is five root six.

It's time to summarise what we've done in our lesson today.

The general rule for multiplying surds is that the square root of x multiplied by y is equal to the square root of x multiplied by the square root of y.

And when dividing, the square root of x divided by y is equal to the square root of x divided by the square root of y.

It is worth bearing mind that sometimes, but not always, it is more efficient to simplify before we multiply or divide with surds.

Well done.

You've put in a lot of effort today and you've done really well.

I look forward to seeing you for our next lesson.