Lesson video

Hi, my name is Mr. Clasper and today we're going to learn how to divide powers.

Let's have a look at this example.

Two to the power of seven divided by two to the power of three.

This is equivalent to two to the power of seven, all over two to the power of three.

As this fraction also represents a calculation involved in division.

We could also write this, as a calculation.

So the numerator represents a calculation equivalent to two to the power of seven, and the denominator represents a calculation equivalent to two to the power of three.

Likewise, if we look at the denominator, if we multiply this by one, this will not change the value.

So the denominator in this case is still equivalent to two, to the power of three.

Taking this a step further, we could write our new fraction as a product of seven different fractions.

And if we look at this closely, the first three fractions of our calculation all have a value of one.

As two divided by two is equal to one.

Or another way to think about this is that we're thinking about two halves and two halves would equal one whole.

Likewise, if we look at the four fractions at the end of the calculation, two divided by one is equal to two.

So therefore, our calculation could be simplified further.

Now again, if we look at this, what if we multiply by one, it will not change the value.

So this calculation is equivalent to this calculation.

And two multiply by two, multiply by two, multiply by two is equal to two to the power of four.

Therefore two to the power of seven divided by two to the power of three must be equal to two to the power of four.

Let's have a look at the same example in a slightly different way.

We have our equivalent fraction and we know that we can write it this way.

What we can look for now are divisions of two.

So every time I divide two by two, this will give me a value of one and this will cancel out.

So two divided by two would give me a value of one, two divided by two would give me another value of one and two divided by two would give me yet another value of one.

This leaves us with two multiply by two, multiply by two, multiply by two.

Therefore we get our answer again of two to the power of four.

Let's have a look at this example.

Three to the power of six divided by three to the power of four.

First of all, this could be written as three to the power of six, all over three to the power of four.

Then we could rewrite the numerator and denominator to represent a calculation that is equivalent, and we can start the same processes again.

So every time we divide three by three, we will get a value of one or one whole.

So three divided by three would give me one, three divided by three would give me another value of one, three divided by three would give me one and three divided by three would give me another value of one.

This leaves a calculation of three multiplied by three.

Therefore my answer is three to the power of two.

Now the rule that we can use, only divide powers, is that when the bases are the same, we can subtract our powers.

So the example in the box is that if we have X to the power of A divided by X to the power of B, that would give us X to the power of A minus B.

So we can subtract the second power from the first power to simplify our answer.

And again notice that the base numbers will always remain the same.

Let's try this example, seven to the power of 13 divided by seven to the power of four.

I know that my base is going to be seven because the base numbers always stay the same and I need to calculate 13, subtract four, which would give me a value of nine.

Therefore this would be seven to the power of nine.

In this example, my base is negative five.

So that means my answer will also have a base of negative five and I must subtract my powers.

So 20 subtract four would give me a value of 16.

So my final answer would be negative five to the power of 16.

Notice that we didn't divide the powers.

That is a common error.

We don't divide our powers, we subtract them.

Our last example, 19 to the power of eight over 19 to the power of seven.

This is exactly the same as 19 to the power of eight divided by 19 to the power of seven.

Therefore my base will be 19 and I need to calculate eight, subtract seven, which would leave me with one.

Therefore my answer should be 19 to the power of one, but this is also equivalent to 19.

Here's a question for you to try.

Choosing A, B, C, or D.

Can you identify the correct answer? Pause the video to complete your task and resume once you're finished.

The correct answer is A, E to the power of 15.

So remember we need to subtract our powers.

So 20 subtract five would give me 15.

Here are some questions for you to try.

Pause the video, to complete your task and resume once you're finished.

And here are your solutions.

So if we take a look at question one D first of all, if we have a look at this one, we've got a base of negative 19, and we have powers of nine and one.

So in this case, we're going to subtract one from nine, which would give us a new power of eight, giving us our final answer of negative 19 to the power of eight.

And if you take a look at question two, having a look to see, we've got three to the power of 19, and we're dividing this by three, but remember that is three to the power of one.

So again, we're going to subtract one from our power of 19, which gives us three to the power of 18.

Let's have a look at this example.

We need to make sure that we're careful when we're subtracting, as subtraction is not commutative.

Therefore it makes a difference depending on which order you've placed your numbers in your calculation.

For this calculation, we're going to need to do five, subtract nine, not nine, subtract five, as this would give us the incorrect answer.

So five subtract nine would give us negative four.

So that means our final answer is seven to the power of negative four.

Let's have a look at this example.

13 to the power of eight, all over 13 to the power of negative three.

Again, we need to calculate eight, subtract negative three.

When we calculate this, we get a value of positive 11.

So remember when we subtract a negative value, the new value will always increase.

So our final answer is 13 to the power of 11.

Here are some questions for you to try.

Pause the video, to complete your task and resume once you're finished.

And here are the answers.

So if we take a look at part A, this is false.

We need to calculate two subtract five, not five subtract two.

This is because subtraction is not commutative.

So two subtract five will give us a value of negative three, not three.

So we get our final answer of two to the power of negative three.

Likewise, for part B, we needed to calculate four subtract eight, which would give us a power of negative four.

It also looks like this attempt has been made by dividing the powers, which is also incorrect.

So remember, we need to subtract our powers when we're applying this law.

The other two examples are true.

Before we move further, we need to recap on another rule.

So when we multiply powers, this is where we can add our powers together.

So given the example of five to the power of seven, multiplied by five to the power of four, our new answer, will have a base of five and to get our new power we need to add these together.

So seven plus four will give me 11.

So we get a final answer of five to the power of 11.

Let's try this example.

The first thing we need to do is to simplify the numerator.

So using our multiplication rule, first of all, we can add our powers together.

So our base will still be eight.

And when we add our powers, we get seven.

Therefore this fraction is equivalent to eight, to the power of seven divided by eight to the power of four.

Now we could use our division rule and we can subtract our powers.

So again, the base of our answer will be eight, and we can subtract our powers of seven subtract four will give me three.

So my final answer is eight to the power of three.

Let's have a look at one more example.

This time we need to simplify our denominator.

So again, using our multiplication rule, we can add our powers together and keep our base the same.

So this fraction is equivalent to seven, to the power of four, all over seven to the power of 14.

Now we can use our division rule.

So we need to calculate four subtract 14, which would give me negative 10.

So remember we're not calculating 14 subtract four, as this will be incorrect.

So our new power is negative 10 and our base remains the same.

So the final answer is seven to the power of negative 10.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

And here are your solutions.

So if we take a look at, four A, we can apply our multiplication law for the numerator.

So the numerator will be equivalent to five to the power of nine, as we can add those powers together.

That means we're going to calculate five to the power of nine divided by five to the power of three, which gives us our five to the power of six.

For part B, notice that the solution was given us five to the power of one, but that's also equivalent to five.

And of those is perfectly acceptable.

And for part C, we need to be careful with our negative numbers.

So the numerator would be five to the power of one, and the denominator will be five to the power of negative one.

So that means we're going to calculate five to the power of one divided by five to the power of negative one.

So the calculation involved will be one subtract negative one, which will give us positive two as a power.

Here's our final question.

Pause the video to complete your task and resume once you're finished.

And here is your final solution.

We needed to calculate the width of the given rectangle.

Now, what we need to know is the fact that if the length multiplied by the width would give us the area, then that logically means that the area divided by the length would give us the width.

So if we calculate 17 to the power of eight, divided by 17 to the power of five, that would leave us with our width, which in this case was 17 to the power of three.

So our final answer is 17 to the power of three kilometres.

And it's only kilometres because it's a length.

So don't put kilometres squared.

We're dealing with length, so it has to be kilometres.

And that brings us to the end of our lesson.

I hope you're feeling a lot more confident with dividing powers, and I will hopefully see you soon.