Lesson video

Hi there, my name is Ms. Lambell.

You've made a really good decision to decide to join me today to do some maths.

Come on then.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is laws of indices.

And in this lesson, we will focus on multiplication.

And this is within the unit, arithmetic procedures, index laws.

By the end of this lesson, you will be able to use the laws of indices to multiply two powers where the bases are the same.

Power is a new word that we are going to be using in today's lesson.

Do you know what we mean by the word power? 16 is the fourth power of 2.

Alternatively, this could be written as 2 with an exponent of 4, and this is read as 2 to the power of 4.

Other key words that we will be using in today's lesson are exponential form.

This is when a number multiplied by itself multiple times can be written more simply in exponential form.

The exponent is the number positioned above and to the right of the base value.

It indicates repeated multiplication.

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

A numerical coefficient is a constant multiplier of the variables in a term.

An operation is commutative if the values it is operating on can be written in either order without changing the calculation.

An operation is associative if a repeated application of the operation produces the same result regardless of how pairs of values are grouped.

All of these words we will need to be familiar with during today's lesson.

If you feel you need to recap any of them, pause the video and just read back through them carefully.

Today's lesson is in three separate learning cycles.

In the first one, we will explore multiplication of powers, and that will be a fairly quick and swift learning cycle.

In learning cycle two, we will look at a multiplication law for powers.

And in the final learning cycle, we will look at changing the base.

Let's get going with that first one.

So we're going to explore multiplication of powers.

Here, we have Laura and Jacob.

Laura says, "I've been thinking about exponents.

They represent a repeated multiplication." Jacob's response is, "Yes Laura, they do.

So, what were you thinking?" Laura says, "When I was doing my homework on prime factors, I collected together all the powers with the same base." Jacob's like, "Yes, and what?" "Well," Laura says.

"2 cubed multiplied by 2 to the power of 5 is a repeated multiplication of 2 three times multiplied by a repeated multiplication of 2 five times." "Which is a repeated multiplication of 2 eight times in total, which is 2 to the power of 8." That's Jacob's response.

Laura says, "Yes, the new exponent is the sum of the original exponents." "Of course, it must be," Jacob says.

You are increasing the total number of repeated multiplication.

What do you think? Yes, it's true.

We'll take a look at this and see how it works.

Now we will show that 2 cubed multiplied by 2 to the power of 5 equals 2 to the power of 8.

That's the one that Laura referred to.

Well, 2 cubed is a repeated multiplication of 2 three times.

So 2 multiplied by 2, multiplied by 2.

2 to the power of 5 is a repeated multiplication of 2 five times.

2 multiplied by 2, multiplied by 2, multiplied by 2, multiplied by 2.

We then take those and we multiply those together, and then we can write it back in exponential form.

So we can see that I have a repeated multiplication of 2 eight times, which is 2 to the power of 8.

So it certainly looks like Laura was right that we add the original exponents.

What about if I've got a base that is algebraic? Show that x to the power of 5 multiplied by x squared is equal to x to the power of 7.

x to the power of 5 is a repeated multiplication of x five times.

And x squared is a repeated multiplication of x two times.

We are multiplying those two together, and then we can write it back in exponential form.

It's a repeated multiplication of x seven times, giving us x to the power of 7.

The sum of 5 and 2 is 7.

Now you are gonna have a go at this one for me.

Show that y cubed multiplied by y squared is equal to y to the power of 5.

Pause the video and then come back when you are ready.

Now, y cubed is y multiplied by y, multiplied by y.

y squared is y multiplied by y.

Multiply those two together, we end up with y multiplied by y, multiplied by y, multiplied by y, multiplied by y, and then we'll put that back into exponential form, which is y to the power of 5.

Now you are ready to do this task.

I'd like you to show that each of those things are equal.

Pause the video and then when you're ready, come back.

I told you it was gonna be a swift learning cycle, didn't I? Good luck and I'll see you in a moment.

Now we can check those answers.

Question number one, should have a repeated multiplication of 5 four times, a repeated multiplication of 5 six times.

So overall, that's a repeated multiplication of 5 10 times.

Question two, a repeated multiplication of 11 twice, a repeated multiplication of 11 three times, which is a repeated multiplication of 11 five times in total.

Question three, repeated multiplication of a three times, a repeated multiplication of a four times, which is a repeated multiplication of a seven times.

Question four, I've got a repeated multiplication of b twice, then a repeated multiplication of b three times, and a repeated multiplication of b five times, which is a repeated multiplication of b 10 times in total.

And question number five, c to the power of 3 is c multiplied by c, multiplied by c.

c obviously is just c, and c squared is c multiplied by c.

If I multiply all three of those together, I end up with a repeated multiplication of c six times.

Let's move on then to using the multiplication law for powers.

A generalised form of the multiplication law for powers is, if I've got a to the power of m multiplied by a to the power of n, then I can write that as a to the power of m plus n.

So the sum of the original exponents.

So for example, 4 cubed multiplied by 4 to the power of 8 is equal to 4 to the power of 3 add 8, which is 4 to the power of 11.

I find the sum of the original exponents.

I can do that also with negative exponents.

5 to the power of negative 3 multiplied by 5 to the power of 7 is 5 to the power of 4.

I can do it with more than two numbers written in index form.

7 to the power of negative 2 multiplied by 7 to the power of negative 5, multiplied by 7 is 7 to the power of 4.

Just remember that 7, we could write it with an exponent of 1.

We can also apply it to algebra.

x squared multiplied by x to the power of negative 3 is x to the power of negative 1.

And it also applies if we've got fractional or decimal exponents.

y to the power of 3 over 2 multiplied by y to the power of 0.

25.

And then I've given you the answer there, either as a decimal or as a fraction.

In order to use this law, however, remember, the bases must be the same.

In general, when multiplying powers with the same base, you can add the exponents.

Simplify 8 to the power of 5 multiplied by 8 to the power of negative 9, giving our answer in index form.

We know that the multiplication law says that we can sum the original exponents.

The sum of 5 and negative 9 is negative 4.

The answer is 8 to the power of negative 4, and that's in index form.

The index is the negative 4, the exponent.

Your turn now.

I'd like you to simplify 12 to the power of 9 multiplied by 12 to the power of negative 2, and give your answer in index form.

Pause the video and then come back when you are ready.

What did you get? So we needed to sum the original two exponents.

So 9 add negative 2, which gives me 12 to the power of 7.

Simplify e to the power of 4 multiplied by e, multiplied by e squared, and give your answer in index form.

Laura says, "This is e to the power of 6." Is Laura correct? What did you decide? Is she correct? No, she is not.

What mistake do you think Laura has made? She has not included the exponent of 1 for the e, and that's a really common mistake.

So remember, any number or any letter has an exponent of 1, we just don't write it.

The answer should have been e to the power of 7.

We add the original exponents of 4, 1, and 2.

Now let's take a look at this one.

We're gonna simplify 3a to the power of 5 multiplied by 4a to the power of negative 6.

Using the associative and commutative laws, we can rearrange the expression.

I'm going to rearrange it so that I have my numbers, my coefficients at the beginning, and then my powers of the a at the end.

Now I'm going to work on my numbers.

3 multiplied by 4 is 12.

And here, we can apply the multiplication law for powers.

We add the original exponents.

And 5 add negative 6 is negative 1.

So it's going to be multiplied by a to the power of negative 1.

But we know that we can leave out that multiplication symbol when we have a coefficient and an algebraic term.

So the correct answer is 12a to the power of negative 1.

Let's take a look at another one.

5a squared b to the power of negative 2 multiplied by negative 4a to the power of negative 1 b to the power of negative 3.

Again, we will rewrite and rearrange the expression using the associative and commutative laws.

Meaning, we will have all of the numbers at the beginning, then we'll separate out the as and the bs.

So I've got the numbers, and then the powers of a and powers of b.

5 multiplied by negative 4 is negative 20.

And now, I can use the multiplication law for powers because I'm multiplying powers with the same base.

a squared multiplied by a to the power of negative 1.

We're going to sum the original exponents.

That gives me a to the power of 1, but remember, we are just going to write that as a.

Now let's take a look at the bs.

b to the power of negative 2 multiplied by b to the power of negative 3, sum negative 2 and negative 3 is b to the power of negative 5.

Question said, "Simplify," so we need to write it as simply as we possibly can.

So we can omit the multiplication symbols.

The answer was negative 20ab to the power of negative 5.

One more together, and then I know you'll be ready to give one of these a go independently.

Let's rearrange it.

Make sure that we've paired up the numbers.

The powers of a and powers of b.

Negative 2 multiplied by negative 4 is 8.

Now we will look at the multiplication law for powers.

We're gonna sum the exponents.

4 add negative 6 is negative 2.

And then we'll do the same with b.

Negative 2 add 8 is 6, so b to the power of 6.

And then rewrite omitting the multiplication symbols.

So it's 8a to the power of negative 2 b to the power of 6.

Now, your turn.

I'd like you to simplify this one.

Pause the video and then when you've got your answer, come back.

Let's check your answer then.

So we rearrange it.

Let's start with the number part.

Negative 3 multiplied by 2 is negative 6.

Now let's consider the powers of a.

a to the power of 5 multiplied by a is a to the power of 6.

Find the sum of 5 and 1.

And then b, we find the sum of negative 3 and 4, which is just b because, remember, that would give us an exponent of 1 which we don't need to write.

And then we rewrite it omitting the multiplication symbols.

Negative 6a to the power of 6 b.

Now I'd like you to have a go at task B.

So the first question, I'd like you please to use that multiplication law for powers to write these as a single power.

Pause the video and then when you're ready, come back.

And question number two, I'd like you to fill in the missing numbers this time.

Superb work.

And now onto question three, you're gonna find the product of the following.

Well done.

Now for those answers.

One, a to the power of 10, b to the power of 3, c to the power of negative 9, d to the power of 4, e and f to the power of 0.

45 or f to the power of 9 over 20.

And on question two, the missing number in the first one was 3.

The second one, negative 7, the third one was 3, the fourth one was 3, the fifth one was 13, and the last one was 1/5.

And then finally, the answers to question three.

A, 10a to the power of 7 b to the power of 11.

B, negative 24a to the power of negative 5 b to the power of 4.

C, 24ab to the power of negative 5.

D, 12a to the power of 7 b cubed add 8a to the power of 6 b to the power of 11.

And E, negative 30a to the power of 10 b add 12ab to the power of negative 3.

Now if I've gone through those too quickly, then you can pause the video and check your answers, and then rejoin me when you are ready to have a go at the third learning cycle.

And this third learning cycle is where we are going to change the base.

Simplify 4 multiplied by 2 to the power of 7, giving your answer in index form.

Laura says, "We cannot do this as the bases are not the same.

We can only use the multiplication law when the bases are the same." Can you see a way, though, that we could simplify this? Jake says, "Yes Laura, you are right.

But we can write 4 with a base of 2." "Oh yes, Jacob, you are right.

We can." So we are gonna decide now.

We couldn't actually write 4 with a base of 2.

What can 4 be written as then? 4 can be written as what? And remember, the base needs to be 2 so that we can use the multiplication law of powers to simplify this.

4 is written as two to the power of what? That's right.

4 is 2 squared.

This is our expression, and we're going to rewrite that, exchanging the 4 for 2 squared.

Now you can see that we can use the multiplication law for powers and we sum the exponents.

So we end up with 2 squared multiplied by 2 to the power of 7 is 2 to the power of 9.

Let's try this one.

We're gonna simplify this one.

27 can be rewritten as what? This time, the base is going to need to be 3 because otherwise we won't be able to simplify it.

27 is 3 to the power of what? Well done.

It's 3 cubed.

3 cubed is 27.

So take our 27 multiplied by 3 to the negative 2, replace 27 with 3 cubed, and then we can apply the multiplication law for indices where we find the sum of the original exponents.

So the answer to this is 3.

Remember, the sum of 3 and negative 2 is 1, but we do not write that exponent of 1.

What about this one? Now Laura says, "Maybe 8 squared can also be written as a power of 2." So she's noticed that we've got 2 to the negative 1.

In order to simplify, the bases have to be the same.

Can 8 squared be written as a power of 2? It can.

Can you write 8 squared as a power of 2 for me? 8 squared is 64, and that's the same as 2 to the power of 6.

So 8 squared is 2 to the power of 6.

Let's rewrite the expression.

We're going to replace 64 with 2 to the power of 6, and then we're gonna apply the multiplication law for powers.

We're gonna sum negative 1 and 6, and the sum of negative 1 and 6 is 5.

The answer is 2 to the power of 5.

Now a quick check for you before I set you going on your final task for today's lesson.

True or false.

You cannot write 625 multiplied by 5 to the power of negative 3 as a single power of 5.

Is that true or false? And then I'd like you to decide which is the correct justification, is it A or B? The bases are not the same, or 625 can be written as a power of 5.

Pause the video, make your decision with a justification, and then come back when you are ready to check your answer.

And you decided.

It was false.

The bases are not the same.

That's true.

But actually, 625 can be written as a power of 5.

625 is 5 to the power of 4.

And so we could simplify this, we could replace 625 with 5 to the power of 4, and then apply multiplication law for powers, sum the exponents, and the sum of 4 and negative 3 is 1.

Remember, we don't write that exponent of 1.

I know you didn't.

Now you are ready to have a go at this task.

So you're gonna simplify each of these by writing them as a single power.

Pause the video, and then when you are ready, come back and we'll check those answers.

Great work.

Here we go then.

Here are your answers.

A is five to the power of 6.

B, 4 to the power of 8.

C, 5 squared.

D, 3 to the power of negative 1.

E, 2 to the power of 12.

F, 5 to the power of negative 3.

G, 2 to the power of 11.

And H is 4.

I've also shown you there how I've worked those out.

So 25, for example, in the first one, I've written as 5 squared.

So if you've made any errors, I suggest you pause the video and then you take a look, and see if you can see where you went wrong.

During this lesson then, we've looked at the generalised form for the multiplication law for powers.

And that is, as long as the bases are the same, then we can add the exponents.

So for example, y to the 3 over 2 multiplied by y to the power of 0.

25 is y to the power of 1.

75 or y to the power of 7 over 4.

Remember, we must make sure that they have the same base in order to use that law.

But it is possible to simplify expressions such as 2 to the power of minus 1 multiplied by 64 by rewriting 64 using the same base.

And in this case, that was 64 equals 2 to the power of 6.

Well done on today's lesson.

Enjoyed working alongside you, and I hope you decide to join me again really soon to do some maths.

Take care and goodbye.