# Lesson video

In progress...

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

Let's have a look at this example.

Write 81, as a single power of three.

Now what this question is asking us to do, is to find something which is equivalent to the value of 81, which has a base of three.

So in other words, we need three to the power of something, which is equal to 81.

Now unless you already know what this value is, the best way to do this is to investigate.

So we should know that three to the power of two is nine.

So our answer cannot be three to the power of two.

Three to the power three, is equal to 27.

So that's three multiplied by three multiply by three.

And again, this doesn't have a value of 81 so this can't be our correct answer.

If we calculate three to the power of four, that is equal to 81.

Therefore our final response for this question will be, three to the power of four.

Let's have a look at this example.

If two to the power of n is equal to 32, find the value of n.

So again, what we need to do is to find the value of the power, that will give us an equivalent value of 32 when we have a base of two.

So this is essentially our problem.

So once again, two to the power of two will be equal to four.

So that's not our correct answer.

Two to the power of three, is eight.

This is not the correct answer, we need a value of 32.

If we try two to the power of four, that will give us a value of 16.

However, if we try to the power of five that gives us a value of 32.

And the question states, find the value of n.

So in this question, n is equal to five.

Here's some questions for you to try.

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

If we take a look at question one e, just be careful with this one.

So we needed to make a value of two.

So as a power of two, this will be two to the power of one, which is equal to the number two.

And if I take question two, if I have a look at question two c, we're looking for four to the power of p is equal to 256.

So I need to replace that letter P with a power, that will give us something that is equivalent to 256.

This will be the number four, as four multiply by four, multiplying by four, multiply by four, would give us 256.

And the last example d, our base is actually a decimal.

So if we think carefully, if five to the power of three, is 125, that will help us to achieve our decimal of 0.

125 So if we replace the base with 0.

5, 0.

5 to the power of three would give us 0.

125 Let's try this example.

Write 16 to the power of five as a single power of two.

Now again, what this question is asking, is for a value, with a base of two, and the power, which is equivalent to 16 to the power of five.

What we can do is if we look at our base of 16, this is equivalent to, two multiply by multiply by two, multiply by two, or two to the power of four.

So if 16 is equal to two to the power of four, that must mean that 16 to the power of five, is equal to two to the power of four raised to the power of five.

Once we're at this point, we can use our power of power rule.

So we can keep our base the same, and we can multiply our powers.

So our final response, would be two to the power of 20 as four multiplied by five would give us 20.

Let's try this example.

Write 81, to the power of six as a single power of three.

So like the last example, we need, to have a response which has a base of three, and a power which is also equivalent to 81 to the power of six.

So if we look at our original base of 81, this is equivalent to three to the power of four.

That means that if 81 is equal to three to the power of four, then 81 raise to the power of six must be equal to three to the power of four, or raised to the power of six.

And again, using our power of power rule, we can keep our bases the same.

So our base will be three, and we multiply the powers given.

So four multiplied by six would give us 24.

So my final response would be three, to the power of 24.

Here are some questions for you to try.

Click resume once you're finished.

Let's take a look at question three.

So, the base 16 can be written, as two to the power of four.

That means we can raise two to the power of four to the power of three instead giving us our answer of two to the power of 12.

Please take a look at question four.

81 can be written as three to the power of four, so I'm going to replace my base of 81 with a base of three to the power of four, and then raise that to a power of three which gives me three to the power of 12 as my final response.

And for question five, 32 can be written as two to the power of five, therefore this calculation can be rewritten as two to the power of five, multiply by two, or multiply by two to the power of one, which would give me a final answer of two to the power of six.

Let's have a look at this example.

Write eight multiplied by 16 as a single power of two.

So once again, a single power of two means that our final response must have a base of two, and the power.

So we have a calculation of eight multiply by 16.

However, eight is equivalent to two to the power three, and 16, is equivalent to two to the power of four.

And using our multiplication rule at this point, we can keep our base the same, and we can add our powers together.

So this would give us a final response, of two to the power of seven.

Let's try one more.

Write 25 multiply by 625, as a single power of five.

So once again, our final response must have a base of five, and the power.

So looking at our calculation, 25 can be written as five to the power of two.

And 625 can be written as five to the power of four.

And once again, using our multiplication rule, we can add our powers together and keep our base the same, so that gives us a final response of five, to the power of six.

Here are some questions for you to try.

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

So if we take a look at question b, 27 can be rewritten as three to the power of three, and nine can be rewritten as three to the power of two.

So multiplying these two we would get three to the power of five.

Similarly, with question d 243, can be written as three to the power of five, and 81 can be written as three to the power of four.

So again, multiplying these two together instead we would get three to the power of nine and for e, 27 to the power of two, 27 is equivalent to three to the power of three.

So if we raise this, to the power of two instead, we would get three to the power of six using our power of power rule.

Write 16 to the power three, or raised to the power of four, as a single power of two.

So once again, our final response would have a base of two, and a power.

Let's take a look at the number 16.

So from previous examples, we should know that 16 is equal to two to the power of four.

So that means, instead of calculating 16 to the power three, or raised to the power of four, we can replace 16 with two to the power of four as it is equivalent in value.

My next step is to simplify my first pair of brackets.

Using my power of power rule, I can multiply my powers together, so four multiplied by three would give me 12, giving me a result of two to the power of 12.

I can now rewrite this once again, and raise two to the power of 12 to the power of four.

And again, using my power of power rule, I can multiply these powers, so 12 multiplied by four would give me 48, giving me a final answer of two to the power of 48.

Here are some questions for you to try.

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

So we take a look at question seven.

Eight squared, can also be written, as two to the power of three, to the power of two.

That means that this can be rewritten as two to the power of six, so my bracket is equivalent to two to the power of six, that if I raised this to the power of three, that would give me my final answer of two to the power of 18.

If we look at question eight, which two cards are equal? So four to the power of three, is equal to, two squared to the power of three, which is equal to two to the power of six.

And if we look carefully at the last card, this is also equivalent to two to the power of six.

And for part B, which two cards have a product equal to two to the power of nine, that will be 16, and 32.

This is because 16 can be written as two to the power of four, and 32 can be written as two to the power of five and multiplying these together would give us a value of two to the power of nine.

And that brings us to the end of our lesson.

I hope you're feeling more confident with powers, Why not try the exit quiz at the end, to show off your new skills.

I'll hopefully see you soon.