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

Before we begin, let's have a quick recap with our multiplication rule.

So, when we multiply with the same base, we can add our powers together.

That will be useful in a second.

Let's have a look at our first example.

We're going to simplify five to the power of two, all raised to the power of three.

Now, another way to write this could be five to the power of two, multiply by five to the power of two, multiply by five to the power of two.

And using my rule, I can add my powers together, and keep my base the same.

So, two plus two plus two is six, or three, lots of two is six.

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

Let's simplify three to the power of five, all raised to the power of four.

Now, we could rewrite this, like this.

And again, using my multiplication rule, I could add my powers together.

So, five plus five plus five plus five, would give me a value of 20.

So, therefore, the final answer should be three to the power of 20.

Alternatively, as I have four lots of five, I could have calculated four multiplied by five to get 20 as well.

This links us to our new rule.

So, when we raise a power to another power, we can multiply the two powers which are involved.

Let's have a look at an example.

Five to the power of two all raised to the power of four, our base will still be five, and we can multiply our two powers.

So, two multiply by four would give me eight.

So, my final answer is five to the power of eight.

In this example, my base would still be seven.

And I can multiply my two powers, which should give me 45.

So, my final answer is seven to the power of 45.

And for my last example, my base will remain the same.

So, my base will still be 13, and again, I can multiply my powers.

So, seven multiplied by four would give me 28.

So, my final answer should be 13 to the power of 28.

This rule will still apply when we have negative powers.

So, for this example, my base will still be two.

However, the calculation I need to carry out would be two multiplied by negative four, and this would give me a value of negative eight.

So my final answer should be two to the power of negative eight.

In this example, my base would still be nine, and this time I need to calculate negative four multiply by five, which will give me a value of negative 20.

So, my final response is nine to the power of negative 20.

And for my last example, my base would still be five, and I need to calculate negative three multiply by negative four.

However, I know that when I multiply two negative values, the answer will be positive.

So, my final answer should be five to the power of positive 12.

Here are some questions for you to try.

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

So, we just need to remember to apply our new rule when we have brackets involved, so, if we look at one, c, our base stays the same, and we multiply our powers, so, six multiply by eight will give us a power of 48.

Moving further down to question two, we need to be careful with negative numbers here, so, if we look at two b, we need to calculate negative two multiplied by seven.

So, a negative multiplying by a positive will always give us a negative result.

So, our answer is five to the power of negative 14.

If we look at part c, we have to multiply negative three by negative four, So, when we multiply two negative values, we will always get a positive value, so, our final answer is five to the power of 12.

For the next examples, we're going to combine some rules.

So, we're going to use the power of power rule, and our multiplication rule.

Let's have a look at this example, we need to simplify this.

So, looking at the term on the left, using our power of power rule, this will be equivalent to two to the power of eight, as my base remains the same, and I can multiply my powers.

So, that means this expression is equivalent to two to the power of eight multiply by two to the power of seven.

Now, to simplify fully, I can now use my multiplication rule.

So, when I multiply powers with the same base, I can add the powers together.

Therefore, eight plus seven will give me 15, and my final answer should be two to the power of 15.

Here are some questions for you to try.

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

Let's take a look at three a, this statement is false.

So, if we look carefully, we should have a base of three, not nine.

So, remember when we're applying this rule, we need to make sure that if our bases are the same, our solution also has the same base.

And if we take a look at three d, we can see that this statement is false, so, if we have 13 to the power of 10, and we raise this to the power of 0.

5, we still need to multiply our powers.

So, 10 multiply by 0.

5, or multiply by a half would be five, therefore, the correct answer should be 13 to the power of five.

Let's have a look at this example, two to the power of two, all raised to the power of a, is equal to two to the power of 12.

So, in this example, we need to find the value of a.

So, using our rule, we know that when we multiply our two powers together, they must equal 12.

So, two lots of a must be equal to 12.

Therefore, a must be equal to six.

And this makes sense because if we had two to the power of two, all raised to the power of six, this will be equivalent to two to the power of 12.

Let's try this example.

Three a to the power of negative five is equal to three to the power of 20.

Now, again we know that our two powers on the left hand side must multiply together and give us a value of 20.

So, we know that negative five multiplied by a or negative five lots of a is equal to 20.

And if we divide both sides of this equation by negative five, we would find that a would have to be negative four.

And again, we can check this because if it was three to the power of negative four all raised to the power of negative five, negative four multiply by negative five would give us a value of positive 20.

Making it correct.

Here are some questions for you to try.

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

So, looking at part a, we can see the correct answer is 19, as the base numbers always remain the same when we're applying this rule.

For part b, we needed a number multiply by three to get six, that gives us two.

For part c, we need a number multiplied by negative five to give us positive 15.

So, that means the value of m must be negative in order to give us a positive value when we multiply it.

That means that the value was negative three.

And for part d, we need nine multiplied by something, which gives us 4.

5 Well, if we look carefully, we should be able to spot that 4.

5 is half of nine.

So, to multiply nine by something to get 4.

5, that could be 0.

5.

If you've written m is equal to 1/2, that's also fine as they're both equivalent.

Here's some questions to try.

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

For question five, we need to find the volume of this cube.

Now, we should know that if the length is seven to the power of nine, then all of its length are seven to the power of nine.

We should also know that to find the volume of a cube, we need to multiply the height by the length by the width.

But, because this is a cube and these lengths are the same, we can apply our new rule.

So, if we calculate seven to the power of nine, and then raise that to the power of three, this will give us our volume.

So, applying our rule, nine multiply by three would give us 27.

So, the volume of our cube is seven to the power of 27 millimetres cubed.

Don't forget your units as well, they're very important.

And for question six, we're given a show that question, so, we need to show that six to the power of two, multiply by six to the power of eight, all over six to the power of three, all raised to the power of four is equal to six to the power of 28.

Now, to show this, I would just calculate this.

So, if we start with the numerator, we've got six to the power of two multiply by six to the power of eight, which is 16 to the power of 10.

We then need to calculate six to the power of 10 divided by six to the power of three, which gives us six to the power of seven, So, that means our bracket is equivalent to six to the power of seven, then we need to calculate six to the power of seven, all to the power of four.

So, applying our new rule again, if we multiply those powers, we get six to the power of 28, thus showing that those two are equivalent.

And that concludes our lesson on calculating powers of powers.

I hope you're feeling more confident with this.

And I'll hopefully see you soon.