Lesson video

Hi everyone, my name is Miss Qu, and I'm really excited to be learning with you today.

It's going to be an interesting and fun lesson.

We'll be building on some previous knowledge as well as looking at some keywords that you may or may not know.

It might be easy or hard in some places, but I'll be here to help.

I'm really excited to be learning with you, so we can learn together.

In today's lesson, from the unit properties of number, factors, multiple squares, and cubes, we'll be looking at power notation, and by the end of the lesson you'll be able to understand and use correct notation for positive integer exponents.

So let's have a look at these keywords.

Well numbers that have been multiplied by themselves a repeated number of times can be expressed using a base and an exponent.

For example, two multiplied by two multiplied by two.

It can be written as two with this little floating three.

Now that little floating three indicates that it's an integer exponent, and the exponent identifies how many times the base has been multiplied by itself.

And the two in this example is the base, and the base is the number or the term that has been multiplied by itself.

Now there's quite a few keywords here, but don't worry we'll be going through these quite a lot in the lesson.

So our lesson today will be broken into two parts.

The first part will be looking at notation, and the second part will be looking at simplifying using power notation.

So let's start by looking at notation.

Notation and pronunciation is so important in mathematics, and we use bases and exponents to identify the power of a number.

This is because we can represent really big numbers in such a small, concise way.

For example, let's look at seven to the power of four, and look at what it means.

Well, we have a base of seven and an exponent of four, but what does it mean? Well, we pronounce it as seven to the power of four, and it means seven multiplied by seven multiplied by seven multiplied by seven.

In other words, the base number is multiplied by itself four times.

So let's have a look at another example.

I want you to look at this, and I want you to think what is the base, what is the exponent, how do you think you pronounce it, and what does it mean? Well, hopefully you've spotted the base is five, the exponent is six, and we pronounce it as five to the power of six.

And it means five multiplied by five multiplied by five multiplied by five multiplied by five multiplied by five.

So you can see why we use bases and exponents as it's much easier than writing the calculation.

Numbers with an exponent of two or three are special, and they have two types of pronunciation.

So let me show you an example.

Here, we have an example where we have a base of eight and the exponent is two.

We know we pronounce it as eight to the power of two, and we know it means eight multiplied by eight.

But because the exponent is two, we can also say eight squared.

So it's the base squared when the exponent is two.

Now I'm going to show you when we have the same base, but the exponent is three.

We can pronounce it as eight to the power of three.

And we know it means eight multiplied by eight multiplied by eight, but we can also pronounce it as eight cubed.

So it's important to know there are two different ways to pronounce a number when the exponent is two or three.

So what I've done here is draw a table, and I'd like you to fill in the table and the first one is done for you.

All I've done is identify a power, and from here you need to know what the base is, the exponent, its meaning, and then identify the pronunciation.

I've done the first for you, so four with that little floating two means we have a base of four, an exponent of two, it means four multiplied by four, and we pronounce it as four squared.

You can also pronounce it as four to the power of two, but I want you to keep to that easy pronunciation when we have an exponent of two or three.

See if you can give it a go, and press pause if you need.

So let's see how you got on.

So our power is that 13 with that little floating two.

So what does it mean? Well, it means the base is 13, our exponent is two, so it means 13 multiplied by 13.

Pronounce it as 13 squared.

Next we have the five with a little floating three.

Well, the base is a five, the exponent is a three, and it means five multiplied by five multiplied by five, as we have a base of five and an exponent of three.

We pronounce it as five cubed.

10 with that little floating four means we have a base of 10, an exponent of four, and it means 10 multiplied by 10 multiplied by 10 multiplied by 10.

We pronounce it as 10 to the power of four.

Next we have three with that little floating five.

Well, the base is three, the exponent is five, and it means three multiplied by three multiplied by three multiplied by three multiplied by three.

And we pronounce it as three to the power of five.

Next, we have 11 with that little floating five, it means we have a base of 11, the exponent is five, and it means 11 times 11 times 11 times 11 times 11.

And we pronounce it as 11 to the power of five.

So you can see why we use bases and exponents, as we don't want to be writing down that long calculation each time.

It simplifies it and makes things much easier.

Now let's do another check question.

Does five to the power of three means five times three? True or false? Well, hopefully you've spotted it's false, but let's see if we can justify why it's false.

Well, is it because five to the power of three means five add five add five? Or is it because five to the power of three means five times five times five? What do you think? Well, using our definition of a base and exponent, I'm hoping that you've identified five to the power of three means five multiplied by five multiplied by five.

Well done.

Let's have a look at another check question.

Here, we're asked to put these numbers in ascending order, smallest to largest.

We have three add three add three, five times three, four times three, five squared, five add five, and four cubed.

So it's important that you're able to calculate the power using the base and the exponent.

See if you can give it a go and press pause if you need.

Well done.

So let's start identifying our answers first.

Well three add three add three is nine.

Five times three is 15.

Four times three is 12.

Five squared.

Well, we know that means five multiplied by five, which is 25.

Five add five is 10.

And four cubed means four multiplied by four multiplied by four, which is 64.

So putting these in order, starting from the smallest, we have three add three add three, five add five, four times three, five times three, five squared, and four cubed.

A huge well done if you got that one right now.

Now let's move on to your task question.

Question one states that Jacob has spilt ink all over his math notes, and we're asked to identify what was under the spilt ink.

So the first part, we don't know what the power is, we don't know what the base is, we don't know what the exponent is, and we don't know how to pronounce it, but we do know it means 12 multiplied by 12.

The second part, well, we don't know what the power is, we don't know what the base is, we don't know what the exponent is.

The pronunciation starts with nine to the power of something, but we know it means nine multiplied by nine multiplied by nine multiplied by nine.

And the last part states four to the power of something.

Well, we know the base is four.

We don't know what the exponent is, we're not sure in the pronunciation, but we know the final answer is 64.

See if you can give this a go and press pause if you need.

Well done.

So let's move on to the second question.

Here, we're asked to put these numbers in ascending order, starting from the smallest to the largest.

Remember to calculate the value of the power using your knowledge on base and exponent.

See if you can give it a go and press pause if you need.

So now let's have a look at question three.

Question three shows us some numbers, and they've been put in ascending order, but we're missing the exponent base or both.

Now we have to use the numbers 2, 2, 3, 4, 8 and 10, and put these into the box, so our numbers are in ascending order.

See if you can also find a different way to complete part A.

Press pause if you need more time.

Well done.

So let's move on to our answers.

For question one, let's identify what's underneath these inkblots.

Well, we know for the first part it means 12 times 12.

So that means it has to be 12 squared with a base of 12, exponent of two, and we pronounce it as 12 squared, or 12 to the power of two.

For the second part, we know it means nine times nine times nine times nine.

So this means we have nine to the power of four.

It has a base of nine, the exponent is four, and we pronounce it as nine to the power of four.

For the third part, we know the answer is 64.

So four to the power of what is 64? Hopefully, you would have figured out, it's three.

So the base is four, the exponent is three, and we pronounce it as four cubed.

I would've also accepted four to the power of three.

Well done if you got that one right.

Next one.

To put these numbers in ascending order, we needed to calculate the value first.

So we know nine squared means nine times nine, which is 81.

Two cubed means two times two times two, which is eight.

Nine times two is 18.

10 times four is 40.

10 to the four means 10 times 10 times 10 times 10, which is 10,000.

And two add two add two is six.

So putting them in ascending order smallest to largest, starts with two add two add two, two cubed, nine times two, 10 times four, nine squared, and 10 to the four.

Huge well done, if you've got that one right.

Moving on to question three, we had to use the numbers 2, 2, 3, 4, 8 and 10, and correctly insert them, so we have our numbers in ascending order.

I'm gonna show you some examples, but there are lots of different examples out there.

Firstly, two cubed, five squared, three to the four, 10 squared, and two to the eight.

These numbers are now in ascending order.

There are quite a few different examples out there, so a huge well done if you've got any of these right.

Fantastic work so far.

So let's look at the second part of our lesson.

Simplifying using power notation.

It's so important to simplify using exponents.

We don't want to be writing this huge calculation each time.

So have a little think of how we can use our knowledge on bases and exponents to simplify this calculation.

It can be simplified to two to the power of 10, which is much easier to write.

We have the exponent is 10, and we have the base is two.

As you can see in our calculation, we have 10 twos all multiplied by themselves.

So this is much easier to write.

So using exponents, what do you think this simplifies to? I'm hoping you've spotted that we have two multiplied by two multiplied by two multiplied by two.

So the base is two with an exponent of four, and it's being multiplied by a base of five and an exponent of three.

So we can simplify our calculation even when we have different bases.

It's important to remember it's convention to put the bases in ascending order.

Now I'm going to give you another example.

Three times seven times seven times three times seven times five times five.

What do you think this simplifies to using exponents? I'm just going to rewrite this, so I can clearly see my bases more easily.

I'm going to group all my threes, my fives, and my sevens just to make it a little bit easier.

Now looking at our calculation, I can see I'm going to use a base of three, a base of five, and a base of seven.

So let's put in our exponents.

It would be three squared times five squared times seven cubed.

A huge well done if you got this one right.

Now, let's move on to a check.

Can you simplify the following using exponents? You can see how powerful power notation is because we're making it so much easier for ourselves rather than writing a huge calculation.

See if you can give it a go and press pause if you need.

Well done.

So let's see what you've got.

For A, hopefully you spotted it's two cubed times three to the power of four.

For B, we have two times five squared times seven to the four.

Now you may have written as two to the power of one.

We don't really write two to the power of one, as we're simply indicating there's only one two being multiplied in this calculation.

For C, hopefully you've spotted it's six to the power of four times eight cubed.

And for D, I tried to trick you here and jumbled up the numbers, but it makes no difference because we still have three cubed times four to the five times five.

A huge well done if you've got that one right.

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

Aisha says two, multiplied by five squared can be simplified to 10 squared.

You have to explain why Aisha is incorrect.

See if you can give it a go and press pause if you need.

This is a great question, so let's work out each part.

Two times five squared.

Well, this means two times five times five, which we can work out to be 50.

10 squared is 10 multiplied by 10, which we know is a hundred.

So this explains why she is incorrect.

Well done if you got that one right now.

Let's have a look at what Jacob says.

Jacob says, seven squared times eight squared times seven squared times eight squared can be simplified to seven to the power of four multiplied by eight to the power of four.

Is Jacob correct, and can you explain? Let's break the calculation down.

Seven squared times eight squared times seven squared times eight squared means seven times seven times eight times eight times seven times seven times eight times eight.

And we know this can be rewritten as seven to the four times eight to the four.

As you can see, we have seven repeated four times and eight repeated four times.

Huge well done if you got that one right.

Now let's move on to the task.

Question one wants you to simplify the following, using exponents.

See if you can give it a go and press pause if you need.

Moving on to question two, we have to pair each statement with the correct simplification, two squared multiplied by five to the power of three, three squared multiplied by two to the power of five, two squared multiplied by two squared multiplied by seven to the power of four, and four to the power of seven multiplied by seven to the power of four.

See if you can pair these up with the correct calculation, press pause if you need more time.

Well done, so let's move on.

Question three wants us to write the following exponents and write the expression in words.

See if you can give it a go and press pause if you need.

Well done.

So let's go through our answers.

Question one wants us to simplify the following, using exponents.

For 1A, I'm hoping you would've got two cubed there, multiplied by three to the power of four.

For B, hopefully you've spotted we have four twos all multiplied by each other, so it's two to the four, and then we have six threes all multiplied by each other, so it's multiplied by three to the six.

For question C, don't worry if they're all mixed up, just remember to count how many times you have a repeated base.

Well, hopefully you've counted, you have three twos all multiplied by each other, so it's two to the power of three, multiplied by three to the power of three multiplied by five to the power of four.

Well done if you've got that one right.

Now let's have a look at pronunciation.

We have to pair each statement with the correct simplification.

Two squared, multiplied by five to the power of three.

Hopefully you spot it here.

Three squared multiplied by two to the power of five.

Hopefully you've spotted it here.

Two squared multiplied by two squared multiplied by seven to the power of four, this would be this one.

And four to the power of seven multiplied by seven to the power of four.

This was a great question, massive well done if you got that one right.

So looking at question three, we're asked to simplify using exponents and write the following in words.

So let's have a look at the first part.

We don't want to be writing it as two multiplied by two multiplied by two multiplied by two multiplied by nine multiplied by nine.

That won't be using exponents, and it's not a nice simplified way.

Using exponents, it's going to be two to the power of four multiplied by nine squared.

Writing that as a sentence or in words, you can see two to the power of four multiplied by nine squared.

Next, you don't want to be writing two multiplied by two multiplied by three multiplied by three multiplied by five multiplied by five.

Woof! We want to be writing it using exponents.

So it would be written as two squared times three squared times five squared.

Next, we certainly don't want to be writing it as two times three times, seven times two times three times seven times two.

Using exponents, it would be two cubed times three squared times seven squared.

Great work if you've got this one right.

So in summary, a number that has been multiplied by themselves a repeated number of times can be expressed using a base and an exponent.

Simplifying using base and exponents presents mathematics so much more simply and clearly, as well as saving so much time.

We know the base represents the number or the term that's been multiplied by itself, and the integer exponent identifies how many times that base number has been multiplied by itself.

Fantastic work, it's been great learning with you today.