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Hello, my name is Mr. Clasper, and today we're going to learn how to multiply with powers.

Before we start to multiply powers, we need to understand what this notation means.

So given the example of five to the power of four, the number five represents what we call the base.

And the number four represents the index or the power.

So for example, if we have five to the power of two, five is the base and two is the power.

Now the power indicates how many times the number five is used in a multiplication.

So as the power is two, that means we use the number five twice in a multiplication.

If we have five to the power of three, because the power is now three, this means that we have a multiplication involving the number five three times.

And if we have five to the power of four, we have a multiplication involving the number five, four times.

Let's have a look at this example.

Simplify five to the power of two, multiplied by five to the power of four.

Well, using what we've just learned, we could write five to the power of two as five multiplied by five.

And we could write five to the power of four as five multiply by five, multiply by five, multiply by five.

So altogether, we have a multiplication using a base of five, six times.

So this means our simplification will become five to the power of six.

Let's try this example.

Simplify seven to the power three, multiply by seven to the power of five.

While seven to the power of three could be written as seven multiply by seven, multiply by seven.

And seven to the power of five, could be written like this.

Altogether, we have a multiplication involving a base of seven, eight times.

Therefore this can be simplified to seven to the power of eight.

Let's have a look at the general rule.

So the general rule is that when we multiply powers together with the same base, we get a simplified answer with the same base, and we can add those powers together.

So looking at the example of six to the power of four multiplied by six to the power of five, this would simplify to six to the power of nine, because four plus five would give us nine.

This example, our simplified solution would have a base of 17.

And when we add 13 and six, we get a new power of 19.

So our final answer will be 17 to the power of 19.

And in this example, we have three to the power of 12 multiplied by three to the power of one.

We just don't write the one in this example.

So therefore, our base will be three.

And we need to add 12 and one together, which would give us three to the power of 13.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you are finished.

And here are the answers.

Let's take a look at question two B.

So when we look at two B, it's important to note that the second four actually has a power of one.

We just don't tend to write that power of one.

So we need to add our powers together which would give us four to the power of four for this example.

And looking further ahead, we can see for two D, the base is negative three but again, all of our other rules stay the same.

So the base will stay the same.

So the base continues to be negative three.

And again, we use our rule, which is to add our powers, which means that we get a power of 100 for this example.

Let's have a look at some different examples.

This time we have negative powers, but our rule remains the same.

So our base will be the same, and we can add our powers.

In this case, our powers are nine and negative five.

So we need to calculate nine plus negative five which would give us a value of positive four.

Therefore my final answer would be six to the power of four.

Looking at this example, I have a base of two.

So my new answer will also have a base of two, and I need to add my powers together.

So negative three plus negative five would give me a value of negative eight.

Therefore my correct answer would be two to the power of negative eight.

Let's take a look at this example.

This time I have decimal powers, but my rule remains the same.

My base will still be seven, and I need to add my powers together.

0.

3 plus 2.

9 will give me 3.

2.

Therefore my final answer would be seven to the power of 3.

2.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you are finished.

And here are your answers.

So looking at question three, part A was false.

So we need to add our powers negative three plus seven would give us five to the power of four.

So our base has to stay the same.

If we look at part B, this example is true.

So negative three plus four is one, but we don't have to write our power of one.

For part C, we have four plus negative four, which is zero.

But that means that our base stays the same and our power is zero.

So the correct answer should be negative 13 to the power of zero.

And for part D, we've got negative three plus negative eight, which should give us negative 11.

So our correct answer was 27 to the power of negative 11.

Similarly with question four, just be careful.

So apply the same rule, just this time involving decimals.

So for the last example four C, we have negative 0.

36 plus 1.

36, which would give us a value of one.

Therefore our final answer is 213 to the power of one or just 213.

Let's have a look at some more examples.

17 to the power of two multiplied by eight to the power of seven is equal to 17 to the power of nine.

And I'm asked to find the value of A.

Well to do this, we know that our bases always remain the same when we apply our power rule.

Therefore, in this example, A would have to be 17.

Let's try this example.

Six to the power of seven multiplied by six to the power of A must be equal to six to the power of 10.

And again, I can apply my law.

So I know that seven plus A must be equal to 10.

Therefore, the value of A must be three and seven plus three is equal to 10.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you are finished.

And here are your answers.

So looking at part A, the correct answer is nine as our base numbers must remain the same.

For part B, we need a number to add to seven that will give us 20.

For part C, again, we need negative five plus something, which will give us negative 10.

So that number has to also be negative five.

And for part D, we need three numbers which have a sum of 15.

But all three numbers must be the same value as well.

So that would give us a value of five, as five plus five plus five would give us 15.

Here is your last question.

Pause the video to complete your task and click resume once you are finished.

And here is the solution.

So for this question, we're asked to find the area of the square.

Well, we know some things about the square.

We know that all of the sides are nine to the power of five.

And we should know that to calculate the area of a square, we need to multiply the base by the height.

So that would mean in this case, we're going to calculate nine to the power of five, multiplied by nine to the power of five, which gives us our solution of nine to the power of 10 millimetres squared.

Don't forget your units, they are very important.

So you need to have millimetres squared.

And that concludes our lesson.

I hope you've enjoyed yourself and I hope you're more confident with multiplying powers.

I'll hopefully see you soon.