Lesson video

Hi there, my name's Ms. Lambell.

You've made such a fantastic choice deciding to join me today to do some maths.

Come on, let's get going.

Welcome to today's lesson.

The title of today's lesson is Laws of Indices.

That's with the division.

This is within the unit, Arithmetic Procedures and Index Laws.

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

Some keywords that we'll be referring to in today's lesson are exponent, coefficient, and power.

A quick recap of those keywords.

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

It indicates a repeated multiplication.

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

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

And 16 is the fourth power of 2.

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

" Today's lesson is in two separate learning cycles.

In the first one, we will explore division of powers.

And in the second one, we will use the division law for powers that we've discovered.

Let's get going with the first one.

So we're going to explore division of powers.

Here we have Andeep and Sam.

Andeep says, "We know there is a multiplication law for powers.

Do you think there is a division law?

" Sam's response is, "There probably is because multiplication and division are inverses of each other.

" What do you think?

Let's explore what happens when we divide powers.

I'm going to simplify 3 to the power of 5 divided by 3 squared.

The first thing we're going to do is we're going to write our division as a fraction, and in expanded form.

We've got 3 X 3, X 3, X 3, X 3, over 3 X 3.

We are now going to rewrite that with a fraction that is equivalent to 1.

We know that 3 X 3 over 3 X 3 is 1.

Now we're going to rewrite that back into exponent form, and we can see that this is 3 to the power of 3.

Do you notice anything?

If you don't, don't worry, we're going to have a look at another example.

We're going to simplify 4 to the power of 7 divided by 4 to the power of 5.

As we did before, the first thing we're going to do is to write our division as a fraction, in expanded form.

We're then going to rewrite with a fraction that is equivalent to 1.

We know that this is equivalent to 1 because the numerator and the denominator are the same.

This leaves us with 4 squared when we write it back in exponent form.

Do you notice anything?

If you noticed something before, does it still work?

If you didn't notice something before, do you now?

You subtract the exponents.

4 to the power of 7 divided by 4 to the power of 5 is 4 squared.

We've subtracted the exponents.

7 subtract 5 is 2.

Now we're going to show that, we're going to show that 2 to the power of 6 divided by 2 cubed equals 2 cubed.

Write the division as a fraction.

Then rewrite with a fraction that is equivalent to 1.

We know that's equivalent to 1, and then rewrite back into exponent form.

We end up with 2 cubed.

We've shown that 2 to the power of 6 divided by 2 cubed is equal to 2 cubed.

Let's do this one together, and then you can have a go at the one on the right hand side on your own.

5 to the power of 7 divided by 5 squared.

We need to show that that is equivalent to 5 to the power of 5.

We're going to write the division as a fraction.

Then we rewrite with a fraction that is equivalent to 1.

We know that 5 X 5 over 5 X 5 is 1.

So we're left with 5 to the power of 5.

We've now shown that 5 to the power of 7 divided by 5 squared is 5 to the power of 5.

Now you can have a go at this one on your own.

Show that 8 to the power of 5 divided by 8 cubed is equal to 8 squared.

Pause the video, and then when you're ready, you can come back and we'll check that answer.

Super work.

Well done.

Write your division as a fraction, then write with a fraction that's equivalent to 1, and then we're left with 8 squared.

Now for Task A.

I'd like you to pause the video, give these questions a go, and then when you are ready, come back and we'll check those answers.

Good luck.

Now let's check your answers.

So I'm not gonna read all of this out.

What I'd like you to do now is to pause the video and then check your answers to question one and two.

And then when you've done that, we'll come back and we'll check the rest of the questions.

And here we are, question three and four.

And finally, question five.

Now we'll move on to our second learning cycle.

We're going to use that division law for powers now.

A generalized form of the division law for powers is: a to the power of m divided by a to the power of n is equal to a to the power of m subtract n.

So for example, 4 to the power of 8 divided by 4 to the power of 5, is 4 to the power of 3, because we subtract the 5 from the 8.

5 to the power of negative 3 divided by 5 to the power of 7, we subtract the powers, we end up with 5 to the power of negative 10.

7 to the power of negative 2 divided by 7 to the power of negative 5, subtract the powers.

Now we need to take care here because we're subtracting a negative number, and we end up with 7 to the power of 3.

Remember also, we can use this law with algebraic terms.

The bases are the same, the bases are both x, so the law applies.

x squared divided by x to the power of negative 3 is x to the power of 5.

2 subtract negative 3 is 5.

And also, we can apply the law if we are not working with in integers.

So here, we're working with fractions and we're working with decimals, but the law still applies.

Really important to remember, in order to use this law, the bases must be the same.

In general, then, when dividing powers, we subtract the exponents.

Let's give this one a go.

8 to the power of 5 divided by 8 to the power of negative 3.

We need to simplify our answer and give it in index form.

8 to the power of 5 divided by 8 to the power of negative 3.

We're dividing, the bases are the same, so therefore we are going to subtract the exponents.

5 subtract negative 3 becomes 5 add 3, which is 8 to the power of 8.

Now you are going to have a go at this one.

Simplify 2 to the power of negative 5 divided by 2 to the power of 12, and you're going to give your answer in index form.

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

How did you get on?

Great work.

2 to the power of negative 5 divided by 2 to the power of 12.

The bases are the same, so we subtract the exponents.

Negative 5 subtract 12 is negative 17, 2 to the power of negative 17.

Simplify b to the power of negative 3 divided by b to the power of negative 1.

Pause the video, decide whether you think your answer is A, B, C, or D, and then when you've got your answer, come back and we'll check it.

What did you decide was the correct answer?

Hopefully you said B.

Of course you said B.

Negative 3 subtract negative 1 is negative 3 add 1, which is b is the power of negative 2.

Simplify 24a to power of 6 divided by 8a squared.

We're going to rewrite our division as a fraction, and then we are going to deal with the coefficients, 24 divided by 8, which is 3.

Then we're going to use the division law for powers to deal with the a to the power of 6 and the a squared.

We are going to subtract the powers, 'cause we're dividing, and we end up with 3a to the power of 4.

Notice that 4 came from 6 subtract 2.

Let's try this one.

Simplify 25a-to-the-power-of-negative-2 b-cubed divided by negative-5a b-to-the-power-of-negative-6.

Let's write it as a fraction, and then we're going to deal with each of the parts separately.

25 divided by negative 5 is negative 5.

Now we're going to deal with the powers of a.

a to the power of negative 2 divided by a will be a to the power of negative 2 subtract 1.

Remember, if there's no exponent, it does have an exponent of 1 really, we don't write it.

Now let's simplify that.

Negative 2 subtract 1 is negative 3.

So we've now got 5a-to-the-power-of-negative-three.

Now we'll deal with the powers of b.

We're gonna do 3 subtract negative 6, which is b to the power of 9.

Now we can take a look at this one.

Sometimes it's easier when there are lots of terms, to separate them.

Let's start with the coefficients.

We've got 4 X 6 divided by 3, and that's 8.

Now we'll deal with the powers of x.

For x to the power of negative 3 X x divided by x squared, we're going to add negative 3 and 1 because we're going to use the multiplication law for powers, and then we're gonna 2 because you can use the division law for powers, and we end up with x to the power of negative 4.

And now we look at the powers of y.

y to the power of 5 X y to power of negative 2 divided by y to the power of negative 1.

So we're gonna add 5 and negative 2, and then we're gonna subtract negative 1, which is 5 add negative 2 add 1, which is y to the power of 4.

And then finally, we need to make sure that we combine all of our terms back together.

Our final answer is 8x to the power of negative 4, y to the power of 4.

We'll try this one now.

So same process.

Let's start with the coefficients.

We've got negative 5 X negative 8 divided by 4, and that's 10.

Then the powers of x.

x cubed X x squared divided by x to the power of 4.

We're gonna add the first two exponents and then subtract the third.

We end up with x, and then the powers of y.

Again, we're going to add the first two exponents and then subtract the third exponent, so we end up with y to the power of negative 2, add negative 1, add 5, which is y squared.

And then don't forget, we need to finally combine all of our terms to give us 10xy squared.

Your turn now.

Pause the video, give this one a go, and then when you are ready, come back and we'll check that answer.

How did you get on?

Super, well done.

Again, we start with the coefficients.

8 X negative 4 divided by 16 is negative 2.

Then the powers of x.

x squared X x to the power of 4 divided by x to the power of negative 1, we're gonna add the first two exponents and then subtract the third exponent.

We end up with x to the power of 7.

and now we'll consider the powers of y.

Y to power of negative 2 X y to the power of 5 divided by y to power of 6.

So again, we're gonna add those first two exponents and then subtract that third exponent.

We end up with y to the power of negative 3.

Don't forget, we need to finally combine all three of those terms.

So the final answer is negative 2x to the power of 7, y to the power of negative 3.

Is that what you got?

Of course you did.

Now you can have a go at this task.

You're going to use the division law for indices to simplify the following.

You need to then find the answers in the grid and shade them to reveal a word.

The answers will appear in the grid more than once.

I'm gonna pause the video, and then come back when you've worked out what that word is.

Good luck with this, and I'll be here waiting when you get back.

You can pause the video now.

Well done.

And question number two.

I'd like you to simplify the following.

So again, pause the video, and then when you've got your four answers, come back.

And finally, question number three.

This time you need to fill in the missing numbers.

What numbers are missing in the boxes?

Pause the video, and then come back when you've got those answers.

Well done.

Let's check the answers then.

A: So question 1-A was 2 to the power of 6.

B, 4 to the power of 8.

C, h to the power of 7.

D was 8.

E, 8 to the power of 7.

F, p to the power of negative 3.

G, 4 squared.

H, r to the power of negative 1.

I, 3 to the power of 6.

J, e to the power of 4.

K was f to the power of negative 6.

And L was g.

And the mystery word was index.

Now let's look at the answers to question number two.

A was 4x cubed y to the power of 4.

B was 4x to the power of negative 9 y to the power of 5.

C was 3x squared y to the power of negative 4.

And D was negative 3 xy.

And finally, question number 3-a.

So missing from the first box was a 4, so an exponent of 4, then the coefficient missing was 5, and then in the third box, the missing exponent was 5.

On to B, the missing coefficient was 36, and then the missing exponent for y was 2, and the missing exponent for x was 5.

C, the missing exponent for x in the first box was 5.

The missing coefficient was negative 6.

And in the denominator, the missing exponent of y was 4.

And then D, the missing coefficient was 3.

The missing exponent of x in the numerator was negative 2.

And the missing exponent of x in the denominator was negative 1.

How did you get on with those?

Those last ones were quite challenging, weren't they?

Now we can summarize the learning that we've done during today's lesson.

A generalized form of the division law for powers is a to the power of m divided by a to the power of n is a to the power of m, subtract n as long as the bases are the same.

So remember, the bases must be the same, and if they are, we can subtract the exponents.

And we've got all of those examples there.

Just make sure you take extra care when you're subtracting a negative exponent.

Thank you for joining me today.

You've done fantastically well.

I look forward to seeing you again really soon.

Take care of yourself, and goodbye.