Lesson video

Hi, everyone.

My name is Miss Coo, and today we're looking at arithmetic procedures, index laws.

I really hope you enjoy the lesson.

It's going to be challenging.

It's got lots of fun tasks to do.

I know I'm going to enjoy teaching it, so let's make a start.

Hi everyone, and welcome to today's lesson on checking and securing understanding of roots and integer indices under the unit, arithmetic procedures, index laws.

And by the end of the lesson you'll be able to calculate with roots and integer indices, including estimating the answer.

Now we'll be looking at the keyword index.

You may already know an exponent is a number positioned above to the right of the base value and it indicates repeated multiplication.

An alternative word for exponent is index, and the plural is indices.

For example, two multiply by two multiply by two is written as two to the power of three, where three is the index and two is the base number or term that has been multiplied by itself.

We'll also be looking at the keywords square number and cube number.

Remember, a square number is the product of two repeated integers and a cube number is the product of three repeated integers.

For example, let's have a look at a square number.

Two multiplied by two is four, so that means four is a square number.

Another example would be five multiplied by five is 25.

So 25 is a square number.

Examples of cube numbers would be two multiply by two multiply by two, which is eight.

So eight is the cube number.

Another example would be five multiplied by five multiplied by five is 125, so 125 is a cube number.

This lesson will be broken in two parts.

The first will be reviewing powers and roots, and the second will be estimating square roots.

So let's make a start reviewing powers and roots.

A perfect square, most commonly called a square number, is the product of two repeated integers.

For example, two multiplied by two is four, and we know four is a square number.

Another example would be five multiplied by five is 25 and 25 is the square number.

Now we can write the calculation to find square numbers using the index of two.

For example, two squared, the index is two is equal to four.

Four is the square number.

Five squared, the index is two is 25.

25 still remains the square number.

So knowing the square number is important as it allows you to efficiently work through calculations.

Now what I want you to do is have a look at this table and I want you to fill in all the square numbers up to and including 15 squared.

So you can give it a go.

Press pause for more time.

It's always handy to have these written down somewhere too.

Great work.

Let's see how you got on with these square numbers.

You should have had 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

These are all our square numbers.

Really well done if you worked those out.

If you're not quite sure on these, make sure you jot them down as we'll be referring to them in the lesson.

Now what I want you to do is a quick check question.

Work out the following and press pause for more time.

Great work.

Let's see how you got on.

Five squared, add two squared, add 10.

Well this is 25, add four, add 10, which is 39.

Eight squared, add four squared, subtract 10, that's 64, add 16, subtract 10, which is 70.

100 subtract six squared, subtract eight squared, well, that's 100, subtract 36, subtract 64, which is zero.

And finally, five squared, add 10 squared, divided by four.

Did you spot the priority of operations here? So it's 25, add 100, divide by four, which then is 25, add 25, which is 50.

Well done if you got that one right.

Now let's have a look at a perfect cube.

A perfect cube, more commonly called a cube number, is the product of three repeated integers.

For example, two multiplied by two multiplied by two is eight, and eight is a cube number.

Another example is five multiplied by five multiplied by five, so 125 is our cube number, but we can rewrite this using an index of three because the index of three indicates the repeated multiplication.

In other words, two cubed indicates two multiplied by two multiplied by two.

So two cubed is eight.

Five cubed indicates five multiplied by five multiplied by five, which is 125.

So what we've done here is written the calculation.

So what we've done here is identify our cube numbers using the index of three.

Now, knowing cube numbers is really important as it does allow you to efficiently work through calculations.

So just like before, can you fill in the table showing all these common cube numbers? Press pause, write them down if it helps, and we'll be referring to this later in the lesson.

Great work.

Let's see how you got on.

Well, one cubed is one, two cubed is eight, three cubed is 27, four cubed is 64, five cubed is 125, and 10 cubed is 1,000.

These are common cube numbers, so try and make sure you learn them.

Now what I want you to do is I want you to do this check.

Put the following numbers in the Venn diagram, you'll see we have the Venn diagram illustrating square numbers, cube numbers.

What do you think goes in the middle, and what do you think goes outside of those circles? So you can give it a go.

Press pause one more time.

Well done.

Let's see how you got on.

Well, here are all our square numbers and cube numbers and the numbers which are neither square or cube.

One and 64 you may notice are square and cube numbers.

Well done if you've got this.

Now what I want you to do is use your knowledge on square and cube numbers to work out the following.

See if you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, for a, five cubed add two cubed add nine is 124, add four add nine, which is 138.

Four cubed, add four squared, take away 10 squared is 64, add 16, subtract 100, which is negative 20.

10 cubed divide by two squared, add eight squared, divide by four cubed.

Well, did you spot the priority operations again? Working this out, you should have 1,000 divide by four, add 64 divided by 64, which is 250, add one, which is 251.

Then five squared times two squared, subtract four cubed divide by two squared.

Priority operations again gives us 25 times four, subtract 64, divide by four, which is 100, subtract 16, which gives us 84.

Well done if you got this.

It's important to know the square root of the perfect square is the integer that has been multiplied by itself to give that perfect square number.

In other words, if you're asked to work out the square to 25, the answer is five.

This is because five multiplied by five gave us that perfect square of 25.

The square root of 36 is six.

That's because six squared gives us our 36.

So now we're going to use the correct mathematical notation.

This symbol known as a radical represents the square root.

So the square root of 25 being five is represented as the square root or the radical of 25 is equal to five.

The square root of 36 is equal to six.

Now what I want you to do, I want to see if you can fill in the rest of these calculations.

You're welcome to use the table if you want, but try and use your knowledge on perfect square numbers to work out the answer.

So you can give it a go.

Press pause if you need more time.

Well done.

Hopefully you've got the square root of 49 is seven.

Square root of 81 is nine, square root of 144 is 12, and the square root of 225 is 15.

Well done if you got this.

So now we have an understanding of what the square root means.

Can you work out the following? See if you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Square root of 25 and the square root of four is five, add 12, which is 17.

Square root of 100, subtract the square root of 81, subtract the square root of one is 10, subtract nine, subtract one, which is zero.

Square root of 16, subtract the square root of one is four, subtract one, which is three.

Four squared, add the square root of 25, so that's 16.

Add five, which is 21.

Square root of 64 divided by two squared.

That is eight divided by four, which is two.

Remember our priority operations again.

We have square root of 225 divided by the square to 25, subtract the square root of 64, divide by the square root of four gives us this calculation.

Using our priority operations, you should have an answer of negative one.

Really well done if you got this one right.

Now let's have a look at another check question.

Joe puts this into his calculator, the square root of negative 100, and the output is this.

I want you to explain why.

Press pause for more time.

Well done.

It's because all square numbers are positive.

No real number multiplied by itself gives negative 100.

Really well done if you understood this.

So now we've done square roots.

Let's have a look at cube roots.

The cube root of a perfect cube is the integer that's been multiplied by itself three times to give that perfect cube number.

In other words, the cube root of eight is two, because two multiplied by two multiplied by two gave us our eight.

The cube root of 125 is five because five multiplied by five multiplied by five gave us that 125.

We use this symbol to represent the cube roots.

So rewriting our sentence, the cube root of eight is equal to two and the cube root of 125 is five.

So now what I want you to do is fill in the rest of these calculations.

I'll give you a couple of seconds.

Press pause if you need more time.

Well done.

So you should have got the cube root of one is one and the cube root of 27 is three.

So now let's check some understanding.

I want you to work out the following using your knowledge of square and cube roots of perfect square and perfect cube numbers.

See if you can give it a go.

Press pause if you need more time.

Well done.

You should have got these answers.

Massive well done if you did.

And it's so important to know your cube numbers and your square numbers.

So that means you can work out the square root of a perfect square and the cube root of a perfect square.

Well done.

Now let's have a look at another calculation.

Laura inputs this into a calculator.

The cube root of negative 1,000, and Jun says, "It will give an error." I want you to explain why Jun is wrong.

Press pause if you need more time.

Well done.

It's because cube numbers can be positive and negative.

The cube root of a negative number is possible because a negative cubed does give a negative number.

Well done if you got this.

Great work, everybody.

Now it's time for your task.

See if you can give these a go.

Press pause for more time.

Well done.

Let's move on to question two.

This is a great question.

Fill in the number to make the calculation correct.

The number in the square is a square number and the number in the cube is a cube number.

Great question.

See if you can work it out.

Well done.

Let's move on to question three.

Nice little combination of square numbers, cube numbers, cube roots, and square roots.

See if you can give it a go.

Press pause for more time.

Well done.

Let's move on to question four.

Same again, little bit harder.

You've got the square root of five squared, add six squared, add eight squared, subtract two squared.

Look how that radical is stretched all the way across.

That means we have implicit brackets there, so be careful.

For B, the cube root of two cubed times five cubed.

Same again.

Look how that radical is stretched all the way across, so we have those implicit brackets.

For C, the square root of a square root of 16, so you can give that a go.

Well done, everybody.

Let's have a look at the last part of our task.

Given that a to the power of x, where x and a are integers, put a tick in the correct place in the table, so you can give it a go.

Press pause for more time.

Fantastic work, everyone.

These were great questions.

So let's go through our answers.

For question one, you should have these.

Very well done.

Press pause if you need more time to mark.

For question two, you should have had these answers.

Massive well done, same again.

Press pause if you need more time to mark.

Excellent work, everybody.

Let's move on to question three.

You should have had these answers.

Well done.

Press pause if you need more time to mark.

And question four, you should have had these.

A little bit more working out for you to look at.

These were quite tricky questions, so very well done if you got this one right.

Excellent, and let's move on to question five.

Well, if a is even and the index is even, does an even number to an even index, does it evaluate to an even number? Yes, it does.

You can even try with some numbers if you want.

So let's think about the base being odd.

If the base is odd and the index is even, does it always evaluate to an even? No, it's false.

Same again, you can try it if you want.

Five to the power of two.

Five is our odd base.

Two is our even index.

That does not evaluate to an even number.

Next, negative number to an even index always evaluates to a negative, true or false? That is false.

Same again, try and have a think.

Negative four to the power of, let's say an index of two.

That gives me a positive number, and we already know all square numbers are positive.

Any even index will always give us a positive number.

Lastly, a negative number to an odd index gives a negative.

That is true.

Fantastic work, everybody.

So let's have a look at estimating square roots.

On this number line I've drawn the numbers 1, 2, 3, 4, 5.

I want you to place, where does the square to one go, the square root of four, the square root of nine, the square root of 16, and the square of 25 go? Think.

Well, they simply go in these positions because we know the square of one is one, square root of four is two, square root of nine is three, and the square root of 16 is four, and the square root of 25 is five.

So let's see how this helps us because we're going to estimate, where do we think the square root of 10 will lie? Have a little look and have a little think.

Well, it's going to lie somewhere in between the square root of nine and the square root of 16.

In other words, somewhere in between three and four.

Do you think the square root of 10 will be closer to the square root of nine or the square root of 16, and I want you to have a little think why.

Well, it's going to be more towards the square root of nine, and the reason is simple, because 10 is just closer to nine than it is 16.

But I'm going to show you another way of how we know for sure using midpoints.

I'm going to use the midpoint of three and four, which is 3.

5, and now I'm going to calculate what 3.

5 squared actually evaluates to.

A nice nifty way is showing you with this square.

Well, we're gonna look at 3.

5 squared and you can see this is 3.

5 by 3.

5.

If I were just to take this area here and simply move it here, that means my calculation has now become three multiplied by four, and then I have this tiny little square to add on.

So what does that mean? Well, it means three multiplied by four, and this tiny little square is actually 0.

25 because it's a 0.

5 multiplied by 0.

5.

We can also show this by writing 3.

5 multiplied by 3.

5 as three, add on 0.

5, multiply by four, subtract our 0.

5, and then we can expand.

And when we expand we still get our 12.

25 and this will always work, for example, 6.

5 squared.

This is the same as six times seven, add on 0.

25, and then we can quickly work it out to be 42.

25.

What about 9.

5 squared? Do you think you can work this one out? Well, it would be nine times 10 and add on our 0.

25, which is 90.

25.

So how does that help us with the original question? Well, if we know 3.

5 squared is 12.

25, that means this is where the square root of 12.

25 lies.

So that means the root of 10 must be less than that.

So the square root of 10 must be closer to three than four.

So if we were to ask to estimate what the square root of 10 would be, the square root of 10 would be approximately 3.

1 or 3.

2 or 3.

3 or 3.

4.

Really well done, and all because we can quickly estimate the square of a 0.

5 number.

So let's do a quick check.

Which of these is a correct way to calculate 8.

5 all squared? So you can give it a go.

Press pause one more time.

Well, hopefully you spotted it's going to be eight times nine, add on.

25.

This is gonna be really helpful in order to estimate square roots.

Now it's time for a check.

I want you to use a number line if it helps to estimate the value of the square root of three.

See if you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, I do know the square root of three has to be in between the square root of one and the square root of four.

In other words, in between one and two.

If I were to work out 1.

5 squared, remember that cute nifty way, one times two, add on 0.

25, that means it's 2.

25.

So that means I know 2.

25 will be here.

So where do you think the square root of three would be? Well, the square root of three must be greater than the square root of 2.

25.

So that means it must be greater than 1.

5 and less than two.

So an example would be the square root of three is approximately 1.

7.

Now what I want you to do is estimate the square root of 137.

Draw a number line if it helps.

See if you can give it a go.

Press pause one more time.

Well done.

Let's see how you got on.

Well, using my number line, I identified the square roots of these perfect square numbers.

The square root of 121 and the square to 144, which is 11 and 12.

So I know the square root of 137 has to be in between here somewhere.

In other words, the square root of 137 has to be in between 11 and 12.

Now if I work out 11.

5 squared, that's 132.

25.

Now this is less than 137, so that means I know the square root of 137 must be greater than 11.

5 and less than 12.

So one example would be the square root of 137 is around about 11.

6.

Well done if you got this.

Great work, everybody.

Now it's time for your final task.

I want you to estimate to one decimal place the values of these.

So you give it a go, draw a number line if it helps, and press pause for more time.

Well done.

So let's see how you got on.

Well, here are all our answers and some example answers too.

Press pause if you need more time to mark.

Great work, everybody.

So in summary, the square root of the perfect square is the integer that has been multiplied by itself to give that perfect square number.

And square numbers are always positive.

And the cube root of the perfect square is the integer that's been multiplied by itself and then itself again to give that perfect cube number, and cube numbers can be positive or negative.

When estimating the square root of a number, drawing a number line really does help.

And working out those mid points can also help identify a good approximation.

Great work, everybody.

It was wonderful learning with you.

I hope you enjoyed the lesson.