Lesson video

Hi there, my name's Ms. Lambeau.

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'll be able to use the laws of indices to divide two powers where the bases are the same.

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

A quick recap of those key words.

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 two.

Alternatively, this can be written as two with an exponent of four, and this is read as two to the power of four.

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 gonna 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 three to the power of five divided by three 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 three multiplied by three, multiplied by three, multiplied by three, multiplied by three, over three multiplied by three.

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

We know that three multiplied by three over three multiplied by three is one.

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

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 four to the power of seven divided by four to the power of five.

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 one.

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

This leaves us with four 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.

Four to the power seven divided by four to the power five is four squared.

We've subtracted the exponents.

Seven subtract five is two.

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

Write the division as a fraction.

Then rewrite with a fraction that is equivalent to one.

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

We end up with two cubed.

We've shown that two to the power of six divided by two cubed is equal to two 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.

Five to the power of seven divided by five squared.

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

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

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

We know that five multiplied by five over five multiplied by five is one.

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

We've now shown that five to the power of seven divided by five squared is five to the power of five.

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

Show that eight to the power five divided by eight cubed is equal to eight 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 one, and then we're left with eight 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 the video and to 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 generalised 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, four to the power of eight divided by four to the power of five is four to the power of three, because we subtract the five from the eight.

Five to the power of negative three divided by five to the power of seven, we subtract the powers, we end up with five to the power of negative 10.

Seven to the power of negative two divided by seven to the power of negative five, subtract the powers.

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

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 three is X to the power of five.

Two subtract negative three is five.

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.

Eight to the power of five divided by eight to the power of negative three.

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

Eight to the power of five divided by eight to the power of negative three.

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

Five subtract negative three becomes five add three, which is eight to the power of eight.

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

Simplify two to the power of negative five divided by two 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.

Two to the power of negative five divided by two to the power of 12.

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

Negative five subtract 12 is negative 17, two to the power of negative 17.

Simplify B to the power of negative three divided by B to the power of negative one.

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 three subtract negative one is negative three add one, which is B is the power of negative two.

Simplify 24A to power of six 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 eight, which is three.

Then we're going to use the division law for powers to deal with the A to the power of six 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 four.

Notice that four came from six subtract two.

Let's try this one.

Simplify 25A-to-the-power-of-negative-two B-cubed divided by negative-5A B-to-the-power-of-negative-six.

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

25 divided by negative five is negative five.

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

A to the power of negative two divided by A will be A to the power of negative two subtract one.

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

Now let's simplify that.

Negative two subtract one is negative three.

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 three subtract negative six, which is B to the power of nine.

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 four multiplied by six divided by three, and that's eight.

Now we'll deal with the powers of X.

For X to the power of negative three multiplied by X divided by X squared, we're going to add negative three and one because we're going to use the multiplication law for powers, and then we're gonna two because you can use the division law for powers, and we end up with X to the power of negative four.

And now we look at the powers of Y.

Y to the power of five multiplied by Y to power of negative two divided by Y to the power of negative one.

So we're gonna add five and negative two, and then we're gonna subtract negative one, which is five add negative two add one, which is Y to the power of four.

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 four Y to the power of four.

We'll try this one now.

So same process.

Let's start with the coefficients.

We've got negative five multiplied by negative eight divided by four, and that's 10.

Then the powers of X.

X cubed multiplied by X squared divided by X to the power four.

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 two, add negative one, add five, 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.

Eight multiplied by negative four divided by 16 is negative two.

Then the powers of X.

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

We end up with X to the power seven.

And now we'll consider the powers of Y.

Y to power of negative two multiplied by Y to the power of five divided by Y to power of six.

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 three.

Don't forget, we need to finally combine all three of those terms. So the final answer is negative two X to the power of seven Y to the power of negative three.

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 one A was two to the power of six.

B, four to the power of eight.

C, H to the power of seven.

D was eight.

E, eight to the power of seven.

F, P to the power of negative three.

G, four squared.

H, R to the power of negative one.

I, three to the power of six.

J, E to the power of four.

K was F to the power of negative six.

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 four.

B was 4X to the power of negative nine Y to the power of five.

C was 3X squared Y to the power of negative four.

And D was negative three XY.

And finally, question number three A.

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

Onto B.

The missing coefficient was 36, and then the missing exponent for Y was two, and the missing exponent for X was five.

C.

The missing exponent for X in the first box was five.

The missing coefficient was negative six.

And in the denominator, the missing exponent of Y was four.

And then D.

The missing coefficient was three.

The missing exponent of X in the numerator was negative two.

And the missing exponent of X in the denominator was negative one.

How did you get on with those? Those last ones were quite challenging, weren't they? Now we can summarise the learning that we've done during today's lesson.

A generalised 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.